If he were to divorce her, his ability to stay in the country would revert to whatever status he had before marrying her. If, subsequent to marrying her, and before divorcing her, he obtained citizenship by the normal process, he'd be free to stay. In your "novel," you'll want to explore his status to determine why he felt he wanted to marry her. If he already has the legal right to live and work here, then "romance" would be the only reason to marry her. If he is illegal, he'd be obliged at some point to reveal his presence in the country and his marriage to a citizen in order to take advantage of it.
Yes, Immigration and Naturalization do check to see if a married couple are living together. See also the response to one of your other questions.
Any person legally in the country who is not a citizen, whether as a temporary or permanent resident, has to report in January of each year to Immigration. Whether or not he were married would not affect that requirement. They would only have to appear before Immigration as a couple at such time as they attempted to establish him as a permanent resident based upon his marriage to a citizen. At that time, the two people are questioned separately in the attempt to determine whether or not they actually are a "genuine" married couple--the point of the questioning is to assure the Immigration Service that they are "legitimately" married, as opposed to a business deal in which she marries him simply to afford him the opportunity to remain in the country.
The State in which they live is immaterial to the Immigration Service, it is a Federal agency. He is still obliged to report to the Immigration Service each January, no matter where he resides.
I would suggest that you visit the site maintained by
the Immigration and Naturalization Service. The information i have given you is based on working with foreign students, many of whom were married to American citizens, more than 25 years ago. It is entirely possible that information i have given you has changed.