Is it permissible for the mother to stipulate to the father that the house be registered in her name, so that she agrees to his marriage to another woman?

Marriage may be obligatory if a man fears falling into sin because one wife is not enough to keep him chaste. It is not permissible for a wife to prevent her husband from a second marriage or to demand money in return for it. She must fear Allah and fulfill her husband's rights. As for the husband, he must be gentle with his wife and appreciate her jealousy, reassure her that her rights are preserved, and deal justly between his wives, otherwise, he will face loss in this world and the Hereafter.