Is it permissible for a couple who have been divorced twice, and whose third divorce is a matter of doubt, to live in the same place without conjugal relations, knowing that the wife lives with her family? And what are the limits of their interaction when meeting their children?

A divorced wife can be subject to another divorce if she is within her waiting period (iddah), because a revocable divorcee is considered like a wife. If you divorced your wife twice while she was still under your marital bond, both divorces are effective. As for the third divorce, if the doubt about its occurrence is due to lack of documentation, then it is considered to have occurred. However, if the doubt is about uttering it or the fulfillment of the oath (hanth), then it is not binding on you, because the original state is the continuation of the marital bond.

If the wife's waiting period ends after the second divorce, she becomes irrevocably separated from you, and she is not permissible for you except with a new marriage contract. It is not permissible to live with her in the same house unless she has an independent room with its own private entrance and exit, and independent facilities.

Interaction with her when visiting the children should be like interacting with any non-mahram woman. It is not permissible to meet her except with the legal hijab, and seclusion with her is forbidden. Speaking with her is also forbidden if it involves softening of speech or leads to temptation.