What is the ruling of Islamic law on a wife who slandered her husband's relative and caused a rupture in family ties, then went to her family's home and openly declared her deed, and do her rights to a deferred dowry and maintenance fall if she is divorced with proof of her insubordination (nushuz)?

Your wife is sinful for spying, slandering, and speaking with a foreign man, and she must repent. Her right to maintenance does not fall unless she is defiant (nashiz), and her defiance has not been proven by what you mentioned. Rational intervention is advised to reconcile between you, and if reconciliation is not possible, then divorce is better. It is not permissible to leave her in limbo; rather, it is either "holding together in kindness or parting with grace." The divorced woman has her rights, and the husband has the right to supplicate against those who wronged him, but forgiveness is better.