If a man marries a woman from Russia so she can become a citizen, stays married for 2 years, is paid a large sum of money by her family to do so, then divorces her, has he committed a crime?

If you're told by someone... maybe yes or maybe no because many times people just reported illegal aliens or marriage frauds, etc to ICE simply because of personal conflicts, however, if you have solid evidences.. yes.

Well said, yes that is basically the definition of marriage fraud.

But you would be surprised how often there is a neighborhood rumor about "the people next door got married so he/she could get a green card" when one person is obviously an immigrant.

If you have no proof just drop it, I have heard of cases where people who report someone to ICE who is not an illegal immigrant has been sued for harassment.