What is the ruling on a wife becoming engaged to another man while she is still married to her husband and the court has not yet issued a ruling on her Khulʿ (divorce at the wife's instance)? And what are the required procedures to prove this engagement?

It is not permissible to propose to a married woman as long as she is still married to her husband. This is a matter of consensus. Even a woman who has been revocably divorced, it is forbidden to explicitly or implicitly propose to her.

The woman's husband must take action to prevent her marriage to another man, and he can resort to the judiciary. If a khul' (divorce at the wife's instance) case is filed, she is not permitted to marry until the court issues its ruling and her waiting period (iddah) ends.

A woman leaving her husband's house without a legitimate religious reason, and her engaging in an emotional relationship with another person, are both reprehensible acts. The sin is compounded if she is married, due to her dereliction of her husband's rights. Her duty is to protect her husband in herself and his property, in fulfillment of the Almighty's saying: "So righteous women are devoutly obedient, guarding in [the husband's] absence what Allah would have them guard."