Can the custodial mother, who divorced her husband through khulʿ, marry another man while retaining custody of her children and having them continue to reside in their father's house, and is it permissible for the divorced husband to reconcile with her before the iddah period ends?

The mother has a greater right to custody of the children so long as she does not marry and before they reach the age of seven. If they reach seven, there is a difference of opinion as to whether they are joined to their father or given the choice between the parents. What is preferred is considering the child's best interest. The residence of the custodian is incumbent upon whoever is responsible for the maintenance of the child. Khul' is an irrevocable divorce, and if the mother marries, her right to custody is forfeited. Custody then transfers to the children's maternal grandmother according to the majority of scholars. If the children exceed the age of custody after the mother's marriage, they are joined to their father or their 'asabah (agnates).