Is it permissible for the second wife to request ownership of a house in her and her children's names to safeguard their rights, especially given the problems with the first family, the in-laws' hatred towards her, their view of her marriage as an injustice, and the fact that the husband's other properties are in the name of his first family?

A man's writing of some of his property in the name of some of his heirs, if the intention is to transfer ownership to them after his death, is not valid and is considered a forbidden will to an heir. It does not take effect unless the remaining heirs agree to it. The husband must be just in gifts among his children from both families, for the Prophet, peace and blessings be upon him, commanded justice among children. There is no harm for a wife to ask her husband to write some properties for her children from him to achieve justice between them and their siblings. There is also no harm for a wife to ask for something from his properties for herself.