Is it necessary to separate from the wife after discovering her exchange of romantic messages with the sister's husband, knowing that there are children and a fear of the impact of separation on them, and a self-reflection regarding forgiveness and equality?

What you mentioned about your wife's contact and relationship with a foreign man is a sin and an act of disobedience. She must repent and cease doing so. It is forbidden for you to condone her actions, for he who condones depravity in his family is a pimp (dayyūth). It has been narrated in a hadith: "There are three whom Allah, the Almighty, will not look at on the Day of Resurrection: the one who disobeys his parents, the woman who imitates men, and the dayyūth." In another hadith: "A dayyūth will not enter Paradise." If your wife repents sincerely, you should keep her and not hold her accountable for a sin she has repented from. As for you, you should repent to Allah for what you have done in the past, and conceal your own faults.