Is the khulʿ that took place through the court valid, given that I did not utter an explicit divorce pronouncement, nor did I receive any money in exchange for it, and the marriage is still considered valid based on that? And is it permissible for the woman to marry another man in this situation?

It is forbidden for the wife's family to separate her from her husband unless there is a strong reason. It is not permissible for the wife to leave the marital home without her husband's permission, nor is it lawful for her to seek divorce or khul' without a legitimate شرعي reason. If there are legitimate reasons, such as harm or intense dislike, she may request khul', and the religious judge (al-Qadi al-Shar'i) is the authority in this matter. If the reasons are substantiated and the wife brings her case before the judge, the majority of scholars consider it commendable for the husband to accede to her request, and some even consider it obligatory, which is the more probable view if there is a fear for the woman's religion or her well-being. In this case, the judge compels the husband to grant khul' if he refuses. In conclusion, the justification for your wife's request for khul' is determined after the religious judge hears the case. If the judge deems it appropriate to annul the marriage, he has the authority to do so, and the request is executed even in the husband's absence if he refuses to attend, with the wife returning her mahr (dowry) and dropping her other claims.