What is the ruling of Islamic law regarding a wife's request for divorce and her desire for custody of the children and the home, and what are the implications of that?

A woman is not permitted to seek a divorce unless there is a legitimate reason. If she has no justification, she has committed a wrongful act, which is a form of insubordination (nushooz). If she insists on the request without a reason, you have the right to refuse to divorce her until she ransoms herself from you with money. There is no harm in this being in exchange for her waiving custody. Custody is a right for both spouses as long as the marriage exists. If a divorce occurs, custody belongs to the mother unless she remarries or there is an impediment.