Is it permissible to demand that the house in which the mother and children live be registered in their names in exchange for the mother waiving all her legitimate entitlements (Mahr, deferred Mahr, Mut'ah, and post-divorce maintenance)?

It is permissible for the father to deed the house to your mother, whether as an outright gift or in exchange for her deferred dower (mahr mu'akhkhar). If your mother is not his wife, he is not obligated to be fair between her and another wife, if one exists. If he wishes to deed the house to his children, there is no harm if he has no other children; otherwise, he must be fair among them in the gift. However, if the house was built by benefactors, it is your property, and he has no claim to it. You are warned against accusing the father of adultery without four witnesses, as this is slander that incurs punishment. Moreover, you must show filial piety and kindness to your father, no matter his condition.