Is it permissible to request khul' or divorce and child custody in cases of infrequent intercourse (12 times in 10 years), emotional dryness, lack of personal hygiene, and weakness of character on the part of the husband?

It is the wife's right upon her husband that he keep her chaste. If the husband falls short in her right regarding the marital bed, she has the right to seek divorce or khul'. Patience is preferable if no harm results from it, especially with the presence of children. The custody of young children belongs to the mother unless she marries a non-mahram or there is an impediment to custody.