What is the ruling of the Sharia and how should we act concerning a house we inherited from our father who did not register it in our names before his death, and another house belonging to our mother which she wishes to gift to us after having paid off its installments, knowing that our father's name still appears in some of its documents?

If the house is owned by the mother, and its registration in the name of both parents is merely a record and not a gift that fulfills its conditions, then this registration is not considered. The mother is permitted to gift the house to her children, provided that she is fair in giving to them, unless there is a legitimate reason to favor some of them, just as the father is obligated to be fair. As for the grandfather selling the father's house, there is no objection to it if it is done with the consent of the adult, mature heirs. However, it is not permissible to dispose of a minor's share except by their legal guardian and in a way that serves the minor's best interest.