Is a khulʿ valid if it took place without the husband's consent and with a forged document, and what is the ruling on the actions of the wife who considers herself divorced and commits acts that violate Islamic law?

A woman is permitted to seek Khul' (divorce initiated by the wife) from her husband if she dislikes him and fears she will not be able to fulfill God's rights with him. However, Khul' is not permissible without justification, as the Prophet, peace and blessings be upon him, said: "Women who seek Khul' are the hypocrites." The original principle is that Khul' occurs by mutual consent, but if the husband refuses and the wife is harmed, the judge may compel him to accept it. It is recommended for the husband to accept the request for Khul' if there is a valid reason for it. What the wife has stated about Khul' is incorrect, as it has not occurred or been decreed by a judge. As for the wife's actions of engaging in usury, going out, and traveling without permission or a mahram (definition) (unmarriageable kin), these indicate her disobedience (nushuz), poor character, and lack of religious commitment. If reconciliation is not effective, it is preferable to divorce her or accept her request for Khul'.