Is a husband obligated to respond to his wife's and her family's request for divorce? And is he sinful for divorcing her, given her disobedience, repeated requests for divorce, and ill-treatment of him?

A wife's repeated requests for divorce without need, her refusal to live in the marital home, and her arrogance, are considered unlawful 'nushuz' (disobedience/rebellion) that forfeits her legitimate rights. Her family is not permitted to demand her divorce; rather, they must strive to reform her. The husband is advised to be patient with her, try to mend her ways, remind her of Allah and the husband's rights, and provide a good environment for her. If she persists in her 'nushuz', then there is no harm in divorcing her, because divorce is permissible when there is a need for it due to the wife's bad character, or it is recommended when she neglects her duties towards Allah.