What do you call a man who cheats on his wife?

A man who cheats on his wife is an adulterer.

Can a married man live with another man legally?

In a recent judgment, the Allahabad High Court ruled that a married woman moving in with another man without divorcing her spouse cannot claim to be in a live-in relationship and seek legal sanctity later. Their act was against the definition of a live-in relationship defined by law and the Supreme Court.