Is the khulʿ that occurred in this case considered religiously valid, and is the wife still under the husband's marital authority, and what is the religious guidance regarding this matter?

"Khulu‘ is the separation of spouses in exchange for compensation received by the husband. It does not occur except with the husband's acceptance and his direct initiation of the khulu‘. What the mazun (authorized marriage official) did, by effecting khulu‘ upon the husband without his consent, is invalid. Valid khulu‘ requires an offer from the husband or his guardian, and an acceptance from the wife or from whoever undertakes to pay the compensation on her behalf. Based on this, the aforementioned khulu‘ is not valid, and the marital bond between them remains intact. It is not permissible for her to marry another man until a valid divorce or khulu‘ occurs and her ‘iddah (waiting period) ends."