Is her second marriage valid? Is her cohabitation with her ex-husband considered adultery? What should the second husband do?

The divorced woman going out with her ex-husband and cohabiting with him after the expiration of her iddah (waiting period) without a new marriage contract is an reprehensible act for which they must both repent. If sexual intercourse occurs, it is zina (fornication/adultery). Their mutual agreement to reconcile does not make any of this permissible except with a new marriage contract.

Her marriage to another man is not problematic if she is free from impediments. As for her marriage before repentance and purification, there is a difference of opinion; the more prevalent view is that the marriage is valid, and he should keep her.

The child is attributed to the second husband if she gives birth to him six months after the marriage contract. If she gives birth to him in less time, the child is attributed to the first husband if he thought it permissible to have intercourse with her after the iddah and before renewing the contract; otherwise, the child is attributed to his mother.