How do I remain in the house and keep custody of my children

You are asking two questions:   First, even with your husband's name only on the mortgage, you have community property interests in the house.  If you file a petition for divorce, the court will decide who gets the house and child and spousal support and visitation with the children.  Sometimes families stay together for economic reasons, and he takes one bedroom and she takes the other.  You can separate and live in the same house.  Usually, however, that is not tolerable when another person comes between you, i.e., boy friend or girl friend.  Bottomline: it doesn't matter whose name is on the mortgage, it's probably community property and owned by both of you.