Praise be to Allah.
The kitaabi (Jewish or Christian) wife should observe ‘iddah following divorce or death of the husband, just like the Muslim woman.
It says in al-Mawsoo‘ah al-Fiqhiyyah (29/336): The Hanafis, Maalikis, Shaafa‘is and Hanbalis, as well as ath-Thawri and Abu ‘Ubayd, are of the view that the kitaabi or dhimmi woman is required to observe ‘iddah in the event of divorce, annulment of the marriage or death of the husband, just like the Muslim woman, because of the general meaning of the evidence that the ‘iddah is required, and no differentiation is made between them on condition that the husband is Muslim, because the ‘iddah is required and is a duty towards Allah and towards the husband. Allah, may He be exalted, says (interpretation of the meaning):
“O you who believe! When you marry believing women, and then divorce them before you have sexual intercourse with them, no Iddah (divorce prescribed period, see have you to count in respect of themem”
So this is his right, and the rights of humans are also due from the kitaabi and dhimmi wife. So she is required to observe ‘iddah and she should be compelled to do so, because of the rights of the husband and child (if she is pregnant), because she is one of those who are required to fulfil the rights of others. End quote.
Living with a wife who is observing ‘iddah varies according to the type of divorce. If it is a revocable divorce (first or second talaaq), then there is nothing wrong with it, because the revocably divorced wife comes under the same ruling as one who is still a wife.
But if it was the final, irrevocable talaaq, then she is a non-mahram and in that case it is not permissible to look at her or be alone with her. Undoubtedly their staying in the same house will make it difficult to adhere to these shar‘i guidelines of not being alone with her and of her observing hijab in front of him, as she should do in front of all non-mahrams. However, if the house is spacious and it is possible to allocate one part for him, with a separate entrance and separate facilities, (then it may be possible for them to live in the same house). But if they are both in the same house, sharing the same entrance and facilities, then it is very difficult to avoid the forbidden matters mentioned above.
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: The thrice-divorced woman is a non-mahram to the man and is like any other non-mahram woman, so the man does not have the right to be alone with her just as he does not have the right to be alone with any other non-mahram woman; he does not have the right to look at her except what he may look at of any other non-mahram woman.
End quote from al-Fataawa al-Kubra, 3/349
The irrevocably-divorced woman is the one who has been divorced three times.
As for the one who has been divorced once or twice, and her ‘iddah has ended without the husband taking her back, she is not permissible for her husband except with a new marriage contract.
And Allah knows best.