Does the divorce issued by the husband via "Messenger" by saying "you are divorced" take effect, especially since it is the third divorce, and what is the ruling on the wife continuing with the Khul’ (divorce at the instance of the wife) case after that?

If the husband says to his wife, "You are divorced," then the divorce takes effect, and the wife is no longer under his legal authority. She becomes unlawful for him if this was the third divorce. The absence of official documentation does not affect the occurrence of the divorce.

If the wife hears the third divorce [pronouncement] and the husband denies it, she must seek his acknowledgment, or free herself from him through khul’ (divorce at the wife's instance) or other means. It is not permissible for her to obey him in marital relations because she has become unlawful for him.