What is the Shar'i ruling regarding a wife who refuses to pray, prefers listening to music with the help of her father, who sees this as "enjoying youth," and left the marital home because her husband refused her going to parties in immodest clothing, and her father is asking the husband to rent another house for her return, even though the marital home is spacious and each individual in it has their own private entrance?

One who abandons prayer out of denial or completely is an infidel, and their marriage becomes null and void. However, one who prays sometimes and neglects it at other times is not considered an infidel. If your wife has abandoned prayer out of denial or completely, it is not permissible for you to remain with her, as the marriage has become null and void. If she prays sometimes and neglects it at other times, you must advise and admonish her. If she persists in her transgression, it is recommended to divorce her, and it is permissible to make things difficult for her so she may ransom herself from you, for holding onto her diminishes your religion.

What the wife's father is doing by inciting her to disobey you and facilitating transgressions is impermissible, and he will be held accountable for it. The wife does not have the right to rent another house as long as you provide her with independent housing.