What is the ruling on a marriage that was built upon the wife's premarital betrayal, and must she be divorced?

The marriage is valid if the Sharia conditions are met. Previous illicit emotional relationships do not affect its validity, but they are an opening for Satan. Divorcing the wife is not obligatory; rather, it is better to be patient with her and try to reform her. What you mentioned about her arrogance and disobedience indicates her nushuz (rebelliousness). The treatment for a rebellious wife involves admonition, boycotting (in bed), and non-violent striking, followed by arbitration. If reconciliation is not achieved, divorcing a woman of bad character is recommended, unless a greater overriding interest opposes it. If you keep her, strive to raise her according to religion and sound morals by teaching her and having her listen to admonitions.