Is it Islamically permissible for the wife and her family to file a lawsuit demanding compensation, alimony, housing rent, and jewelry, along with making false claims?

What the woman did—insulting, humiliating, rebelling, and making false accusations in court—is haram (definition) (forbidden). As for the residence for custody, the general rule is that housing the child is obligatory upon the father. Regarding the residence of the custodian, there is a difference of opinion, but the fatwa (definition) (religious ruling) that is relied upon is that if the woman has a residence, her child lives with her, and she is not entitled to housing rent from the child's father. However, if she does not have a residence, then it is obligatory upon the child's father to provide her with housing during the period of custody. As for the gifted gold, it became her property the moment she took possession of it, and no one is permitted to reclaim it. It is obligatory upon whoever took it to return it to her, based on the Prophet's (peace and blessings be upon him) saying: "The one who takes back his gift is like a dog that vomits and then returns to its vomit."