Where Should a Thrice-Divorced Wife Observe `Iddah?
Question

1- A woman who has been divorced three times has sons and daughters. Where should she spend her `Iddah? In her family’s house or in the marital home with her sons and daughters? Please note that she wants to spend her ‘iddah with her sons and daughters.
2- There was a wife who was divorced three times, and during her ‘iddah she became sad, and when she met her husband (during the `Iddah), she embraced him and things happened, but no kissing or intercourse took place.
What is the way to expiate for this sin by both parties? Please note that no kissing or intercourse took place.
Praise be to Allah.Understanding Irrevocable Divorce and Its Impact on `Iddah
If a man has divorced his wife irrevocably, she is not entitled to any maintenance or accommodation during the `Iddah , unless she is pregnant.
Types of Irrevocable Divorce
Irrevocable divorce is of two types:

Minor irrevocable divorce, which is when the woman is divorced before consummation of the marriage, and divorce in return for compensation (meaning: in return for money taken by the husband).
Major irrevocable divorce, which means completion of three divorces (Talaqat).

Rights to Maintenance and Accommodation for the Pregnant
Ibn Qudamah (may Allah have mercy on him) said: If a man divorces his wife in such a way that has no right to take her back, then she is not entitled to accommodation or maintenance, unless she is pregnant.
In general, if a man divorces his wife irrevocably, either by a third divorce , or Khul` , or an annulment, and she is pregnant, then she is entitled to maintenance and accommodation, according to scholarly consensus, because Allah says (interpretation of the meaning):
{Lodge them (the divorced women) where you dwell, according to your means, and do not harm them so as to straiten them (that they be obliged to leave your house). And if they are pregnant, then spend on them till they lay down their burden.} [At-Talaq 65:6]
According to some reports of Fatimah bint Qays (may Allah be pleased with her), “You are not entitled to any maintenance unless you are pregnant.” and because the pregnancy is his child, so he is obliged to spend on it, and he cannot spend on him (the fetus) except by spending on her, so it is obligatory, just as it is obligatory to pay for the costs of breastfeeding.
But if she is not pregnant, then there is no maintenance for her.
Hadith Evidence on the `Iddah Location for Irrevocable Divorce

With regard to accommodation, there are two reports, one of which is that she is entitled to that. This is the view of `Umar and his son, and Ibn Mas`ud, `Aishah, the seven jurists of Madinah, Malik and Ash-Shafi`i, because of the verse quoted above.
The second report says that she is not entitled to accommodation or maintenance, and this is the prevalent view of the Madhhab, and it is the view of `Ali, Ibn `Abbas, Jabir, `Ata’, Tawus, Al-Hasan, `Ikrimah, Maymun ibn Mahran, Is-haq, Abu Thawr and Dawud.

Most of the Iraqi jurists said that she has the right to accommodation and maintenance. This is the view of Ibn Shubrumah, Ibn Abu Layla, Ath-Thawri, Al-Hasan ibn Salih, Abu Hanifah and his companions, Al-Batti and Al-`Anbari. (End quote from Al-Mughni, 8/185)
The evidence that the irrevocably divorced woman is not entitled to maintenance or accommodation is the report narrated by Muslim (1480) from Ash-Sha`bi who said: I entered upon Fatimah bint Qays and asked her about the judgement that the Messenger of Allah (blessings and peace of Allah be upon him) passed concerning her, and she said that her husband divorced her irrevocably, and she said: I referred my dispute with him concerning accommodation and maintenance to the Messenger of Allah (blessings and peace of Allah be upon him), and he did not give me any accommodation or maintenance, and he told me to observe my `Iddah in the house of Ibn Umm Maktum.
According to another report also narrated by Muslim, she said: I mentioned that to the Messenger of Allah (blessings and peace of Allah be upon him) and he said: “There is no maintenance or accommodation for you.”
According to a report narrated by Abu Dawud, “There is no maintenance for you unless you are pregnant.”
Ibn `Abd Al-Barr (may Allah have mercy on him) said: But with regard to evidence and what is binding thereof, the view of Ahmad bin Hanbal and those who followed him is more sound and more correct, because if it were obligatory for her to stay in her husband’s house and was something ordained by Allah, then Messenger of Allah (blessings and peace of Allah be upon him) would have obliged her to do that and would not have made her leave her husband’s house to go to the house of Umm Sharik, or to the house of Ibn Umm Maktum.
As it is proven that the Prophet (blessings and peace of Allah be upon him) said to Fatimah bint Qays (may Allah be pleased with her), who had been irrevocably divorced: “There is no accommodation or maintenance for you; rather accommodation and maintenance are for the one who may be taken back,” then what can contradict that? Can it be contradicted except by a similar report from the Prophet (blessings and peace of Allah be upon him), who was the one who explained what Allah meant in His Book? And nothing of that nature has been reported from him (blessings and peace of Allah be upon him). It is well known that he knew better than anyone else about the interpretation of the words of Allah (interpretation of the meaning): {Lodge them (the divorced women) where you dwell}. (End quote from At-Tamhid, 19/151)
Conditions for Staying in the Husband’s Home during `Iddah
If the husband allows her to stay in his house during the `Iddah, then there is nothing wrong with her staying there, on condition that she observe Hijab before him, because by virtue of the irrevocable divorce he has become a stranger to her. But it is better for her to observe the `Iddah in her family’s house, so as to close the door to Fitnah, as mentioned in the question, because the devil may make her appear attractive to him, and make him appear attractive to her, until they do a prohibited action, when before that the Shaytan strove hard to make them unattractive to one another until the  threefold divorce took place: {Then take admonition, O you with eyes (to see).} [Al-Hashr 59:2]
Repentance and Boundaries after a Final Divorce
For an irrevocably divorced woman to embrace her former husband is a prohibited action, as was what happened after that of touching him and so on. What they must do is repent to Allah and regret this evil action.
It is no secret that with the third divorce, they became strangers (non-Mahrams) to one another, so it is not permissible to look or touch, let alone the embracing and what you mentioned.
It is not permissible for him to go back to her until she has married another husband, in a genuine marriage, and not a marriage of convenience aimed at making her permissible for the first husband, then he (the second husband) dies or leaves her.
See the answers below for further clarification:
Where should a revocably-divorced woman observe `Iddah?
Husband and wife must stay in the same house during `Iddah
Can a Revocably-Divorced Woman Go Out during `Iddah?
And Allah knows best.