the craving-practice leads to the origin of feeling

From SN 36.23:

Paṭipadā is placing at the same position of upādāna (clinging), so we can find the answer in the sutta that including clinging, craving, and feeling together, such as the four noble truth in Sutta. See Saṃ. Ma. DhammacakkappavattanaSutta:

"Now this, monks, is the noble truth of stress:1 Birth is stressful, aging is stressful, death is stressful; sorrow, lamentation, pain, distress, & despair are stressful; association with the unbeloved is stressful, separation from the loved is stressful, not getting what is wanted is stressful. In short, the five clinging-aggregates are stressful.
"And this, monks, is the noble truth of the origination of stress: the craving that makes for further becoming — accompanied by passion & delight, relishing now here & now there — i.e., craving for sensual pleasure, craving for becoming, craving for non-becoming.

The feeling is an aggregate, and the craving origins clinging-aggregates, so the practicing of craving origin the feeling aggregate. See Sutta. Ma. U. Mahāpuṇṇamasuttaṃ:

"Monk, these are the five clinging-aggregates, i.e., form as a clinging-aggregate, feeling... perception... fabrications... consciousness as a clinging-aggregate."

Therefore, the last question in SN 36.23 is about the noble eightfold path of the four noble truth:

Q:What’s the origin of feeling? katamā nu kho, bhante, vedanā?
...
A: Feeling originates from contact. Phassasamudayā vedanāsamudayo.
...
Q: What’s the practice that leads to the origin of feeling? katamo vedanāsamudayo, katamā vedanāsamudayagāminī paṭipadā?
...
A: Craving is the practice that leads to the origin of feeling. Taṇhā vedanāsamudayagāminī paṭipadā.
...
Q: What’s the cessation of feeling? Katamo vedanānirodho?
...
A: When contact ceases, feeling ceases. Phassanirodhā vedanānirodho.
...
Q: What’s the practice that leads to the cessation of feeling? katamā vedanānirodhagāminī paṭipadā?
...
A: The practice that leads to the cessation of feelings is simply this
noble eightfold path, that is:
Ayameva ariyo aṭṭhaṅgiko maggo vedanānirodhagāminī paṭipadā,
seyyathidaṃ—
right view, right thought, right speech, right action, right
livelihood, right effort, right mindfulness, and right immersion.
sammādiṭṭhi … pe … sammāsamādhi.
...

The explanation of SN 35.23 Sabba Sutta in ṢaḷāyatanaVagga

0. History base knowledge about wrong and right view: See SarathW's Answer.
Then:

1. Why are these called "The All": the eye & forms, ear & sounds, nose & aromas, tongue & flavors, body & tactile sensations, intellect & ideas?
2. Why someone trying to explain "The All" as being something else would fail?
3. What does "it lies beyond range" mean?

The wrong view has the wrong condition(s). The right view has the right condition(s). The all right condition is the dependent origination, because the conditioner and the conditioned relate each other in balance, nothing left out of the cycle of the condition.
the dependent origination explains "Thus arises this whole mass of dukkha".
Each conditioner of the dependent origination causes the whole conditioned effect. For the example, ... the (Six-)Saḷāyatanas-conditioner causes the whole conditioned Six-Phassa; The Six-Phassa-conditioner cause the conditioned six-vedanā ...
So, if one, who questioned, explains the All, whole, into the dependent origination as the Buddha did, that one is able to explain the whole from the beginning to the end follow to the dependent origination such as the Buddha can explain every view in Sutta. Tī. Sī Brahmajālasuttaṃ.
But if one, who questioned, explains the All, whole, out of the dependent origination, that one would be unable to explain, and furthermore, would feel hard to answer (fail). Why? Because it is beyond that one's abilities, he exactly answers the wrong condition (fail), i.e. God condition the being.
SN KhandhaVagga and SN ṢaḷāyatanaVagga, such as SabbaSutta, come after SN Nidānavagga, the dependent origination because they are the whole elements which cycling in the dependent origination.

Buddhism's Existing Period Counting

The contexts about Buddhism's Existing Period Counting appear as 3 tenses in Aṭṭakathā:
1. The existing enlightenment of the current Buddha's follower (45BBE - about 5,000 BE):
Buddhism's existing period counting of current Buddha according to Vinaya.A. (samanta.3) Bhikkhunikkhandhakavaṇṇanā, An.A. (manoratha.1) Dutiyapamadadivaggavannana:
1st 1000 years after Gotama, Current Buddha, died:

Arahanta with Four Analytical Knowledges still occur=>

2nd 1000 years after GotamaSambuddha died:

Arahanta with Six Higher Knowledges still occur=>
Arahanta with Three Clear Knowledges still occur=>
Arahanta via Insight-only still occur=>>

3rd 1000 years after GotamaSambuddha died:

Anāgāmi with Four Analytical Knowledges still occur=>
Anāgāmi with Six Higher Knowledges still occur=>
Anāgāmi with Three Clear Knowledges still occur=>
Anāgāmi via Insight-only still occur=>>

4th 1000 years after GotamaSambuddha died:

Sakadāgāmi with Four Analytical Knowledges still occur=>
Sakadāgāmi with Six Higher Knowledges still occur=>
Sakadāgāmi with Three Clear Knowledges still occur=>
Sakadāgāmi via Insight-only still occur=>>

Final 1000 years after GotamaSambuddha died:

Sotāpanna with Four Analytical Knowledges still occur=>
Sotāpanna with Six Higher Knowledges still occur=>
Sotāpanna with Three Clear Knowledges still occur=>
Sotāpanna via Insight-only still occurs until the end of final 1000 years.

2. The previous KassapaBuddha (uncountable millions of years ago):
KassapaSamBuddha's, previous Buddha's, Buddhism existing period counting appear in Di.A. (sumangala.3) Sampasadaniyasuttavaṇṇanā; It using the past tense verbs). And Saṃ.A. (sārattha.2) Saddhammappaṭirūpakasuttavaṇṇanā, which connects to that Sampasadaniyasuttavaṇṇanā:
1st 1000 years after Kassapa, previous Buddha, died:

Arahanta with Four Analytical Knowledges still occur=>

2rd 1000 years after Kassapa, previous Buddha, died:

Arahanta with Six Higher Knowledges still occur=>

3th 1000 years after Kassapa, previous Buddha, died:

Arahanta with Three Clear Knowledges still occur=>

4th 1000 years after Kassapa, previous Buddha, died:

Arahanta via Insight-only still occur=>>

Final 1000 years after Kassapa, previous Buddha, died:

Pāṭimokkha memorizer still occurs until the end of final 1000 years.

3. The current Buddha's living time (45 - 0 BBE):
The sequence of existing enlightenment of the Buddha's followers just in current alive buddha's time, not died, appear in Saṃ.A. (sārattha.2) Saddhammappaṭirūpakasuttavaṇṇanā.It's explaining the Buddha's answer to KassapaThera's question in Sutta. Saṃ. Ni. Saddhammappaṭirūpakasutta:

"What is the cause, lord, what is the reason, why before there were fewer training rules and yet more monks established in final gnosis, whereas now there are more training rules and yet fewer monks established in final gnosis?"

At alive GotamaSamBuddha's time after he has enlightened, at the beginning Arahanta with Four Analytical Knowledges still occur. As time went by Arahanta with Six Higher Knowledges still occur Arahanta with Three Clear Knowledges still occur. As time went by Arahanta via Insight-only still occur. As time went by Anāgāmī still occur. As time went by Sakadāgāmi still occur. As time went by Sotāpanna still occur. As time went by Sīpavanta still occur.

---

A short term of all above Buddha's periods are up and down depending on the situation, but it tends to go down in long-term of every period. It's something like this:

Cr. wikipedia

The sequence concept of Theravāda Buddhism

The sequence concept of Theravāda Buddhism, start from sīla to paṭisambhidā, summarized from paṭisambhidāmagga, visuddhimagga, abhidhamma, atthhakathā, and all sāriputta's canons:
Paññā = understand the fact: causes&effects.
Paṭivedha = Paññā.
Ñāṇa = Expert Paññā.
Dassana = View, Understood, Diṭṭhi, Cakkhu.
SammāDiṭṭhi =SammāDassaṇa.
Ñātapariññā = Base of sammādassaṇa.
Tīraṇapariññā = Beginning sammādassaṇa practice.
Pahānapariññā = Advance sammādassaṇa practice.
Visuddhi = The base level which cleared from the extended enemy of the next level's enemy, so it is able to go to the next level.
Sīla = Arising physical & verbal behavior.
SīlaAdhiSikkhā = Physical & verbal behavior's training that supporting CittaVisuddhi.
SīlaVisuddhi = The CittaVisuddhi's base level which cleared from AvītikkamaKilesa, extending from CittaVisuddhi's enemy (PariyuṭṭhānaKilesa), so it is able to go to the next level, CittaVisuddhi.
AvītikkamaKilesa = Present life's arising physical & verbal unwholesome, extending from PariyuṭṭhānaKilesa.
PariyuṭṭhānaKilesa = Present life's arising mental unwholesome, extending from AnusayaKilesa.
AnusayaKilesa = Unwholesome behavior which ready to arise in the name "PariyuṭṭhānaKilesa". It arises as the name "PariyuṭṭhānaKilesa", but it stands as the name "AnusayaKilesa". It is similar to behavior because you can't find your behavior in any organ when you are not doing it, but you know your behavior still available to arise. The arising unwhole behavior is AvītikkamaKilesa/PariyuṭṭhānaKilesa.
Citta = Arising mind's behavior, Arising consciousness's behavior, arising Thought's behavior.
CittaAdhiSikkhā = Thought behavior's training that supporting DiṭṭhiVisuddhi.
CittaVisuddhi = The DiṭṭhiVisuddhi's base level which cleared from PariyuṭṭhānaKilesa, extending from DiṭṭhiVisuddhi's enemy (AttaAnuDiṭṭhiAnusayaKilesa), so it is able to go to the next level, DiṭṭhiVisuddhi.
SammāDassana = Right understood of conditioner and conditioned (four noble truth, dependent origination); diṭṭhivisuddhi + kaṇkhāvitaraṇavisuddhi; nāmarūpaparicchedañāṇa+paccayapariggahañāṇa; Ñātapariññā; Vipassanā Paññā; YathābūtaÑāṇaDassana.
four noble truth, dependent origination = conditioner aggregates and conditioned aggregates.
DiṭṭhiVisuddhi = The DiṭṭhiVisuddhi's base level which cleared from AttaAnuDiṭṭhiAnusayaKilesa, extending from KaṇkhāvitaraṇaVisuddhi's enemy (VisamahetukaDiṭṭhiAnusayaKilesa), so it is able to go to the next level, KaṇkhāvitaraṇaVisuddhi.
Attā (of the wrong view) = The misunderstood concept of conditioner aggregates and conditioned aggregates. This misunderstood concept make ordinary people think "there are some controllable aggregates left to be nihilistic or eternal". But the truth is "there are no any controllable aggregates to be nihilistic or eternal. There are only uncontrollable conditioner aggregates and uncontrollable conditioned aggregates".
AttaAnuDiṭṭhi = The misunderstood behavior about aggregates whether "some/whole aggregates are Attā".
KaṅkhāvitaraṇaVisuddhi = The MaggāmaggaÑāṇaDassanaVisuddhi's base level which cleared from VisamahetukaDiṭṭhiAnusayaKilesa, extending from MaggāmaggaÑāṇaDassanaVisuddhi's enemy (VipassanāUpakKilesa), so it is able to go to the next level, MaggāmaggaÑāṇaDassanaVisuddhi.
Vipassanā = SammāDassaṇa; VipassanāPaññā; Vipassanā; Ñātapariññā; Pajānāti.
Anupassnā = ÑāṇaDassaṇa; VipassanāÑāṇa; Vipassanā Bhāvanā; Vipassanā Meditation; Tīraṇapariññā + Pahānapariññā; Anupassati.
MaggāmaggaÑāṇaDassanaVisuddhi = SammāDassaṇa at cleared ñāṇa level from the extending enemy (the ignorant about the right or wrong practice [maggāmagga;paṭipadā]) of the next level's enemy (the ignorant about the four noble truth), so it is able to go to the next level, to practice watching three characteristics without confusing of the right or wrong practice; BapavaUdayabbayañāṇa.
PaṭipadāÑāṇaDassanaVisuddhi = above ÑāṇaDassana at cleared ñāṇa level from the extending enemy (the ignorant about the four noble truth) of the next level enemy (the ignorant about the four noble truth), so it is able to go to the next level, to stop the ignorant about the four noble truth forever by the complete&perfect watching three characteristics; From Bhaṅgañāṇa to Anulomañāṇa.
ÑāṇaDassanaVisuddhi = above ÑāṇaDassana at cleared ñāṇa level(enlightenment) from the extending enemy (fetters; saṇyojana) of the next level enemy (rebirth; becoming; bhava; paṭisandhiviññāṇa), so it is able to go to the next level, to stop the rebirth as many as the level of the enlightenment.
Paṭisambhidā = ÑāṇaDassanaVisuddhi according to Abhi. Vibhaṅgo ñāṇavatthuvibhaṅga, Sutta. Khu. Paṭisambhidāmaggo mahāvaggo ñāṇakathā mātikā, Paṭisambhidāmaggo's atthakathā, Visuddhimagga Khandhaniddeso. All ariya must enlighten Paṭisambhidā of own meditation. They can analysis everything about own meditation (super mundane jhāna), but they maybe don't know about mundane jhāna, Abhiññā, and the other practitioners' meditation. They enlightened Paṭisambhidā of own meditation only, they may be not the Paṭisambhidappatto.
Tevijjo = Ariya who is an expert, vasī, of jhāna. That expertise increases the power of ÑāṇaDassana to be greater, to be able to recall many past lives and to be able to recall many deaths&rebirths more than the practitioner who is no an expert of jhāna. Normally, the only vipassanā practitioner can recall about 1-2 past lives, but the expert of jhāna can recall more than that. The expert of jhāna spend the lesser time to recall the past lives at the same number of life.
Abhiññā = Special practice of an expert, vasī, of jhāna. Two Abhiññā, Pubbenivāsānussatiñāṇa and Cutūpapatañāṇa, have the similar practice as ÑāṇaDassaṇa, so the expert of jhāna can enlighten as a Tevijjo easier.
Chaḷabhiñño = Ariya who is an expert, vasī, of every Abhiññā.
Paṭisambhidappatto = Chaḷabhiñño who is an expert of own and the others' Paṭisambhidā.
For the example question:

if I become a stream-enterer I will automatically have four analytical knowledge right? Will I also have three clear knowledge and six higher knowledge?

if you become a stream-enterer without being an expert of jhāna&abhiñña&paṭisambhidā, you are a stream-enterer who automatically have only four analytical knowledge right of your meditation. You are only a Sukkhavipassako, but you are not Paṭisambhidappatto, Chaḷabhiñño, Tevijjo.
For the conclusion:

In AN 4.171, what is “reincarnation where the intention of others is effective”?

Per commentary, this passage refers to "devas corrupted by the mind" (Manopadosika-deva), living in the world of the Four Great Kings. When they get angry with one another their mutual anger leads to their mutual deaths.
Here are descriptions of Manopadosika-deva from different sources:
Buddhist Dictionary. Manual of Buddhist Terms & Doctrines (4th Edition) by Ven. Nyanatiloka:

Manopadosika-deva: ‘the celestial beings corruptible by temper’, are a class of devas (q.v.) of the sensuous sphere. “They spend their time in becoming annoyed with one another, and getting into a temper, and thus by being bodily and mentally exhausted, they pass from that world” (D. 1; 24).

The "Dictionary of Pali Proper Names" by G. P. Malalasekera:

A class of devas living in the Cātummahārājika world (DA.i.114; AA.ii.544; MNid.108). They burn continually with envy one against another (Buddhaghosa relates a story illustrating this in DA.i.114) and their hearts become ill disposed and debauched. Their bodies thus become feeble and their minds imbecile, and, as a consequence, they fall from their state (D.i.20; iii.32). The Manopadosikā were among those present at the preaching of the Mahāsamaya Sutta (D.ii.260). They are so called because their minds are debauched by envy (dosassa anudahanatāya) (VibhA.498).

Brahmajāla Sutta (DN 1) translated from the Pali by Bhikkhu Bodhi:

There are, bhikkhus, certain gods called 'corrupted by mind.' These gods contemplate one another with excessive envy. As a consequence their minds becomes corrupted by anger towards one another. When their minds are corrupted by anger, their bodies and minds become exhausted and consequently, they pass away from that plane.

Brahmajāla Sutta (DN 1) translated by Bhikkhu Sujato:

There are gods named ‘malevolent’. They spend too much time gazing at each other, so they grow angry with each other, and their bodies and minds get tired. They pass away from that group of gods.

So, one deva's hatred towards another deva provokes hatred in that deva towards the first, causing a vicious circle, and this very hatred exhausts their spirits until they fall from their godly state.
One deva's hatred is not his own deliberate intention but is merely a reaction to another deva's hatred. Therefore the deva falls from his state and comes to a different state through an "intention of other".
It's interesting how the vicious circle of judgement starts not through one of the devas performing some unwelcomed act to another, but rather through both spending too much time wondering about the other's attitude to oneself.
I suppose this basically means that not all karma is strictly personal. While in the simplest cases sentient beings inherit their own karma, there are cases when they create (bad) karma for one another, and there are cases when a karmic pattern emerges spontaneously from a background of ignorance - not through anyone's fault.
On a side note, per my (Russian) dictionary, attabhāvapaṭilābhe is not "reincarnation" but "a state of personal existence".
by Andrei Volkov @https://buddhism.stackexchange.com/a/30153/10100.

---------------------

I agree with Andrei Volkov explanation, which follows to it's atthakathā. However, for the perfect understood by sutta, not only atthakathā, I write this answer.

1. There is a reincarnation where one’s own intention is effective, not that of others. Atthi, bhikkhave, attabhāvapaṭilābho, yasmiṃ attabhāvapaṭilābhe attasañcetanā kamati, no parasañcetanā.

By atthakathā, above text refers to DN Brahmajālasuttaṃ:

7. ‘And what is the second?

‘There are, brethren, certain gods called the “Debauched by Pleasure” [155]. ‘For ages they pass their time in the pursuit of the laughter and sport of sensual lusts. In consequence thereof their self-possession is corrupted, and through the loss of their self-control they fall from that state [156].

8. ‘Now it might well be, brethren, that some being, on his falling from that state, should come hither. And having come hither he should, as in the last case, become a recluse, and acquire the power of recollecting his last birth, but only his last one.
9. ‘And he would say to himself: “Those gods who are not debauched by pleasure are stedfast, immutable, eternal, of a nature that knows no change, and they will remain so forever and ever. [20] But we-who fell from that state, having lost our self-control through being debauched by pleasure-we have come hither as being impermanent, mutable, limited in duration of life.”

By atthakathā, above text refers to MN kandarakasuttaṃ:

2. There is a reincarnation where the intention of others is effective, not one’s own. Atthi, bhikkhave, attabhāvapaṭilābho, yasmiṃ attabhāvapaṭilābhe parasañcetanā kamati, no attasañcetanā.

Bhikkhus, which person torments others yoked to it? Bhikkhus, here a certain person is a killer of pigs, goats, birds, a hunter of wild animals, a fisherman, a robber, a highway robber, one who keeps a prison, or engaged in any other vicious activity. This person torments others yoked to tormenting others.

3. There is a reincarnation where both one’s own and others’ intentions are effective. Atthi, bhikkhave, attabhāvapaṭilābho, yasmiṃ attabhāvapaṭilābhe attasañcetanā ca kamati parasañcetanā ca.

By atthakathā, above text refers to DN Brahmajālasuttaṃ:

10. ‘And what is the third?

‘There are, brethren, certain gods called "the Debauched in Mind [157].” They burn continually with envy [158] one against another, and being thus irritated, their hearts become ill-disposed towards each other, and being thus debauched, their bodies become feeble, and their minds imbecile. And those gods fall from that state.

11. ‘Now it might well be, brethren, that some [\q 034/] being, on his falling from that state, should come hither; and having become a recluse should ‘as in the other cases, acquire the power of recollecting his last birth, but only his last one.
12. ‘And lie would say to himself: “Those gods who are not debauched in mind do not continually burn with envy against each other, so their hearts do not become evil disposed one towards another, nor their bodies feeble and their minds imbecile. Therefore they fall not from that state; they are stedfast, immutable, eternal, of a nature that knows no change, and they will remain so forever and ever. [21] But we were corrupted in mind, being constantly excited by envy against one another. And being thus envious and corrupt our bodies became feeble, and our minds imbecile, and we fell from that state, and have come hither as Being impermanent, mutable, limited in duration of life.”

‘This, brethren, is the third case.

4. There is a reincarnation where neither one’s own nor others’ intentions are effective. Atthi, bhikkhave, attabhāvapaṭilābho, yasmiṃ attabhāvapaṭilābhe nevattasañcetanā kamati, no parasañcetanā.

By atthakathā, above text refers to DN Dānamahapphalasutta:

"Having given this gift seeking his own profit — with a mind attached [to the reward], seeking to store up for himself, [with the thought], 'I'll enjoy this after death' — on the break-up of the body, after death, he reappears in the company of the Devas of the Four Great Kings. Then, having exhausted that action, that power, that status, that sovereignty, he is a returner, coming back to this world.
"Having given this gift with the thought, 'Giving is good,' on the break-up of the body, after death, he reappears in the company of the Devas of the Thirty-three. Then, having exhausted that action, that power, that status, that sovereignty, he is a returner, coming back to this world.
...

Why I have to link? Theravāda tradition never allow to explain tipitaka without reference, linking. It's weird and not professional. If buddhaghosa wrote something without the reference, he wouldn't be allowed to translate atthakathā back to pāli. So, the link, which I have given, show tipitaka memorizer's thought.

Is suffering dependent on aging & death? If so, how is suffering experienced before & after death?

If death must occur for sorrow, lamentation, pain, distress & despair to come into play; what is the cause for people having sorrow, grief & despair before death, i.e., in the present moment?

If death must occur for sorrow, lamentation, pain, distress & despair to come into play; is sorrow, grief & despair only experienced after death?

Aging & death must come together to show "if you born without unstable, it's ok. But actually, your life is unstable because it's going to be aging & death, it's not ok. Because while you are living, birth+aging+death, you are going to have sorrow, lamentation, pain, distress & despair to come into play aging & death."

Pali context only shows birth causes "aging & death, sorrow, lamentation, pain, distress & despair to come into play". Nothing in pāli context shows aging & death cause "sorrow, lamentation, pain, distress & despair to come into play".

jātipaccayā jarāmaraṇaṃ sokaparidevadukkhadomanassupāyāsā sambhavanti.

So, your both questions are misunderstood pali context. There is nothing in pali context shows "death must occur for sorrow, lamentation, pain, distress & despair to come into play". Pāli only shows birth causes "aging & death, sorrow, lamentation, pain, distress & despair to come into play".

In the other hand by pāli context, the sequence of paṭiccasamuppāda is "birth >> aging&death, sorrow, lamentation, pain, distress & despair to come into play". It is not "birth >> aging >> death >> sorrow, lamentation, pain, distress & despair to come into play".

What does “sandhāvitvā saṃsaritvā” mean in AN 9.12?

1. What does "sattakkhattuparama" mean literally and non-literally?

sattakkhattuparama = not more than seven times.

Why did he include rebirth into his translation?

Because of the context below which come before "sattakkhattuparama" and being on the same topic:

There are these nine people who, dying with something left over, are exempt from hell, the animal realm, and the ghost realm. They’re exempt from places of loss, bad places, the underworld.

Navayime, sāriputta, puggalā saupādisesā kālaṃ kurumānā parimuttā nirayā parimuttā tiracchānayoniyā parimuttā pettivisayā parimuttā apāyaduggativinipātā.

...

With the ending of three fetters, and the weakening of greed, hate, and delusion, they’re a once-returner. They come back to this world once only, then make an end of suffering.

So tiṇṇaṃ saṃyojanānaṃ parikkhayā rāgadosamohānaṃ tanuttā sakadāgāmī hoti, sakideva imaṃ lokaṃ āgantvā dukkhassantaṃ karoti.

...

With the ending of three fetters, they’re a one-seeder. They will be reborn just one time in a human existence, then make an end of suffering.

So tiṇṇaṃ saṃyojanānaṃ parikkhayā ekabījī hoti, ekaṃyeva mānusakaṃ bhavaṃ nibbattetvā dukkhassantaṃ karoti.

...

With the ending of three fetters, they go family to family. They will transmigrate between two or three families and then make an end of suffering.

So tiṇṇaṃ saṃyojanānaṃ parikkhayā kolaṅkolo hoti, dve vā tīṇi vā kulāni sandhāvitvā saṃsaritvā dukkhassantaṃ karoti.

Next question:

2. What is the meaning of the phrase "sandhāvitvā saṃsaritvā" in this quote? How does it relate to "sattakkhattuparama"?

saṃsaritvā is the synonym of sandhāvitvā. sandhāvitvā means transmigrate, transport, move next.

Translation: Person (puggalo) moves next(sandhāvitvā saṃsaritvā) not more than seven times (sattakkhattuparamaṃ) in deva [bhava] and human [bhava] (deve ca manusse ca) and then make the end of suffering (dukkhassantaṃ karoti).

3. Is the phrase "sandhāvitvā saṃsaritvā" used in other suttas, without being connected to "sattakkhattuparama"? What does it mean there? What is the context there?

Comepare with "sandhāvitvā saṃsaritvā" in MN Sandakasutta. It's context means "move to next life (rebirth)":

There are 1.4 million main wombs, and 6,000, and 600. There are 500 deeds, and five, and three. There are deeds and half-deeds. There are 62 paths, 62 sub-eons, six classes of rebirth, and eight stages in a person’s life. There are 4,900 Ājīvaka ascetics, 4,900 wanderers, and 4,900 naked ascetics. There are 2,000 faculties, 3,000 hells, and 36 realms of dust. There are seven percipient embryos, seven non-percipient embryos, and seven embryos without attachments. There are seven gods, seven humans, and seven goblins. There are seven lakes, seven winds, seven cliffs, and 700 cliffs. There are seven dreams and 700 dreams. There are 8.4 million great eons through which the foolish and the astute transmigrate before making an end of suffering.

Cuddasa kho panimāni yonipamukhasatasahassāni saṭṭhi ca satāni cha ca satāni pañca ca kammuno satāni pañca ca kammāni tīṇi ca kammāni, kamme ca aḍḍhakamme ca, dvaṭṭhipaṭipadā, dvaṭṭhantarakappā, chaḷābhijātiyo, aṭṭha purisabhūmiyo, ekūnapaññāsa ājīvakasate, ekūnapaññāsa paribbājakasate, ekūnapaññāsa nāgāvāsasate, vīse indriyasate, tiṃse nirayasate, chattiṃsa rajodhātuyo, satta saññīgabbhā, satta asaññīgabbhā, satta nigaṇṭhigabbhā, satta devā, satta mānusā, satta pesācā, satta sarā, satta pavuṭā, satta papātā, satta papātasatāni, satta supinā, satta supinasatāni, cullāsīti mahākappino satasahassāni, yāni bāle ca paṇḍite ca sandhāvitvā saṃsaritvā dukkhassantaṃ karissanti.

Next question:

4. Can I say that "sandhāvitvā saṃsaritvā" supports the interpretation of "sattakkhattuparama" as "seven lifetimes" or "seven rebirths"?

Actually, it is just a sign of many signs. Actually, the whole context of sutta supports the interpretation of "sattakkhattuparamo" as "person who has seven lifetimes left" or "person who has seven rebirths left". There are many signs to translate like that for the example:

With the ending of three fetters, they’re a one-seeder. They will be reborn just one time in a human existence, then make an end of suffering.

So tiṇṇaṃ saṃyojanānaṃ parikkhayā ekabījī hoti, ekaṃyeva mānusakaṃ bhavaṃ nibbattetvā dukkhassantaṃ karoti.

1. Above quote's link is liking with many other suttas which force the translator to translate "ekaṃyeva bhavaṃ nibbattetvā" as "will be reborn just one time".
2. When the previous context translated "ekaṃyeva bhavaṃ nibbattetvā" as "will be reborn just one time", the next context have to translate follow it, too.

Person (puggalo) moves next(sandhāvitvā saṃsaritvā) not more than seven times (sattakkhattuparamaṃ) in deva [bhava] and human [bhava] (deve ca manusse ca) and then make the end of suffering (dukkhassantaṃ karoti).

I have to add 1 more question to complete this answer:

5. Why the buddha didn't use "nibbattetvā" with all person? Why the buddha used "sandhāvitvā saṃsaritvā" instead?

"Ekabījī (one-seeder)" will be reborn just one time (ekaṃyeva bhavaṃ nibbattetvā), but the next others will reborn multiple times. "sandhāvitvā saṃsaritvā" often use in multiple time rebirths.

Saupādisesasutta:

With the ending of three fetters, they’re a one-seeder. They will be reborn just one time in a human existence, then make an end of suffering.

So tiṇṇaṃ saṃyojanānaṃ parikkhayā ekabījī hoti, ekaṃyeva mānusakaṃ bhavaṃ nibbattetvā dukkhassantaṃ karoti.

...

With the ending of three fetters, they go family to family. They will transmigrate between two or three families and then make an end of suffering.

So tiṇṇaṃ saṃyojanānaṃ parikkhayā kolaṅkolo hoti, dve vā tīṇi vā kulāni sandhāvitvā saṃsaritvā dukkhassantaṃ karoti.

...

With the ending of three fetters, they have at most seven rebirths. They will transmigrate at most seven times among devas and humans and then make an end of suffering.

So tiṇṇaṃ saṃyojanānaṃ parikkhayā sattakkhattuparamo hoti, sattakkhattuparamaṃ deve ca manusse ca sandhāvitvā saṃsaritvā dukkhassantaṃ karoti.

MN Sandakasutta:

There are 1.4 million main wombs, and 6,000, and 600. There are 500 deeds, and five, and three. There are deeds and half-deeds. There are 62 paths, 62 sub-eons, six classes of rebirth, and eight stages in a person’s life. There are 4,900 Ājīvaka ascetics, 4,900 wanderers, and 4,900 naked ascetics. There are 2,000 faculties, 3,000 hells, and 36 realms of dust. There are seven percipient embryos, seven non-percipient embryos, and seven embryos without attachments. There are seven gods, seven humans, and seven goblins. There are seven lakes, seven winds, seven cliffs, and 700 cliffs. There are seven dreams and 700 dreams. There are 8.4 million great eons through which the foolish and the astute transmigrate before making an end of suffering.

Cuddasa kho panimāni yonipamukhasatasahassāni saṭṭhi ca satāni cha ca satāni pañca ca kammuno satāni pañca ca kammāni tīṇi ca kammāni, kamme ca aḍḍhakamme ca, dvaṭṭhipaṭipadā, dvaṭṭhantarakappā, chaḷābhijātiyo, aṭṭha purisabhūmiyo, ekūnapaññāsa ājīvakasate, ekūnapaññāsa paribbājakasate, ekūnapaññāsa nāgāvāsasate, vīse indriyasate, tiṃse nirayasate, chattiṃsa rajodhātuyo, satta saññīgabbhā, satta asaññīgabbhā, satta nigaṇṭhigabbhā, satta devā, satta mānusā, satta pesācā, satta sarā, satta pavuṭā, satta papātā, satta papātasatāni, satta supinā, satta supinasatāni, cullāsīti mahākappino satasahassāni, yāni bāle ca paṇḍite ca sandhāvitvā saṃsaritvā dukkhassantaṃ karissanti.

Buddhist objective is nibbāna, the safest, so the reader need to learn it hard step by step

Buddhist objective is nibbāna, the safest.
It's like the science's, engineering's or computer programming's rule, we can't misunderstand at any level of the process. It's buggy, risk and unsafe.
Therefore, it's very hard to understand Tipitaka and Atthakatha as alike as science's, engineering's or computer programming's rule.
So, if you point your expectation in Buddhism only at the easy part, you will miss the safest part, which elaborated multi aspects of reasons. That missing is buggy, risk and unsafe.
So, concrete exposition of Buddhism is the conclusion like:

1. Vinaya is in VN Parivāra by upāli, the vinaya secretary of buddha.
2. Sutta is in KN Paṭisambhidāmagga by sāriputta, the stand-in teacher of buddha in MN Mahāgosiṅgasālasutta (or alternative in translated Visuddhimagga but if you bias of the supernatural, it's not your choice).

Both above require these to understand:

1. Pāli skill from Kaccāyana.
2. Tipitaka analysis skill from Netti.
3. The discipline of everything from abhidhamma by sāriputta, the stand-in teacher of buddha in MN Mahāgosiṅgasālasutta. Or, saṅgaha by anuruddha, past laṅkā's pontiff.

But if you want to start in Buddhism, you need:

1. Qualified teacher to teach you step by step.
2. Easy sutta to recite and do follow, such as Maṅgalasutta (and all Khuddakapātha), Siṅgālakasutta, and dhammapatha.

The Pali word 'saṅkhārā' in Dependent Origination plural because it is the crossroad.

Dependent origination is the cycle of the happening event. Each event has only one decision. There are 2 saṅkhāra for each decision: wholesome and unwholesome. Wholesome saṅkhāra gives good resultants. Unwholesome saṅkhāra gives bad resultants. So, viññāna to vedanā which are the resultants of saṅkhāra must be singular, because it's just the resultants of one decision. Jāti and jarāmaraṇaṃ are same. Wholesome karmma-bhavo gives good resultants. Unwholesome karmma-bhavo gives bad resultants. So, jāti and jarāmaraṇaṃ must be singular because of the same reason.

Above explanation explained clearly in abhidhamma vibhaṅga paccayākāravibhaṅga and it's atthakathā. It separates each event to singular as "saṅkhāro". That's why no direct explaination in any atthakathā, included path of purification. However, it's used widely in difference context such as in VN.A. Introduction.

And this is the example of what I wrote "it's used widely":

Kusalattika: kusalā dhammā, akusalā dhammā, abyākatā dhammā.

The "wholesome" triad: wholesome phenomena, unwholesome phenomena, not both phenomena

The Path of Purification detailedly exposited the dependent origination follow to 'the "wholesome" triad':

jarāmaraṇaṃ = not both phenomena

jati = not both phenomena

bhavo = not both phenomena

upādānaṃ = unwholesome phenomena

taṇhā = unwholesome phenomena

vedanā = not both phenomena

phasso = not both phenomena

saḷāyatanaṃ = not both phenomena

nāmarūpaṃ = not both phenomena

viññāṇaṃ = not both phenomena

saṅkhārā = 1. wholesome phenomena, 2. unwholesome phenomena

avijjā = unwholesome phenomena

You can see whether saṅkhārā are separated into 2 events.

Above quote is my own conclusion. To read it, the reader needs to understand in advance of chapter XIV, XV, XVI, XVII, and some part of chapter I, III. Also, it requires Abhidhamma knowledge in advance, as well.

Path of Purification, CH. XVII, THE SOIL OF UNDERSTANDING—CONCLUSION, DEPENDENT ORIGINATION, III. Detailed Exposition page 547-608.