Is providing a separate residence for the wife without the full consent of the parents considered an act of disobedience (ʿUqūq)? Does the wife have the right to demand a private dwelling? And is the husband sinful if he argues with his mother when he is certain of her error in some family disputes?

You must provide your wife with independent housing if she requests it, especially given the harm she is experiencing from your family. This is true even if your parents disapprove, for there is no obedience to a creature in disobedience to the Creator. However, try to persuade them. As for arguing with your mother, it is not permissible if it involves raising your voice, but it is permissible to advise her gently, as Allah Almighty says: "So do not say to them, 'uff,' and do not repel them but speak to them a noble word. And lower to them the wing of humility out of mercy."