Is the husband obligated to spend on his children, and does the wife's dowry become forfeited, and is she entitled to child custody; after she violated the marriage conditions and refused to return to the marital home?

Firstly, if the wife becomes disobedient (nashiz) and refuses to return to her husband, she is not entitled to maintenance, unless she is pregnant, in which case she is entitled to maintenance for the pregnancy itself.

Secondly, the maintenance of the children is obligatory upon you, and it includes food, clothing, housing rent, and school expenses. You must be careful that the mother does not seize the maintenance for herself, and ensure that it is spent on the children.

Thirdly, the deferred mahr (dowry) is not forfeited by the wife's disobedience. If there is no reason from the husband for seeking a divorce, he may refuse to divorce her and compel her to initiate a khul' (divorce at the wife's instance), in which case she redeems herself from him with the deferred mahr or more or less, according to the agreement, based on the Almighty's saying: "And it is not lawful for you to take anything of what you have given them, unless both fear that they will not be able to keep the limits [set by] Allah. But if you fear that they will not be able to keep the limits of Allah, then there is no blame upon either of them concerning that by which she ransoms herself."

Fourthly, it is not permissible for the wife to resort to deceit to take the children. If a divorce or khul' occurs, custody is determined based on the children's age and the condition of the parents. A woman who drinks alcohol is not fit for custody. You must strive to rescue your children from her authority by every possible means.