Is a wife's leaving her husband's home without his permission, due to his repeated physical abuse of her and their children, and the police threatening to withdraw custody of the children from them if the problems recur, truly forbidden? And what is the optimal solution to this problem?

It is not permissible for a husband to beat his wife severely. Rather, Islamic law has permitted non-severe beating upon the fear of insubordination (nushuz), and this only after admonition and boycotting. This beating should be with a siwak stick or similar, such that it leaves no mark and breaks no bone, and it should not be motivated by revenge. It is not permissible for a wife to leave or travel without her husband's permission, unless there is a justification for it, such as severe beating and fear for her safety. If the husband cannot control himself when angry, the wife may reside in a separate house in the same city, or with her family if possible, to reduce risks and enable the husband to see his children. If the husband continues his aggression, the wife is permitted to seek divorce (talaq (definition)) or khul'. If she agrees to stay, she must give him his rights while avoiding the danger of him taking the children by maintaining a separate residence.