nanafish will have had her reasons for whatever happened. The phrase 'I guess it was an affair' suggests that it wasn't a clear-cut 'cheating' arrangement, but even they sometimes happen for a reason, and those on the outside can't know what they would do in the circumstances unless they have lived them.
As for the predicament the OP is asking about, I would be unhappy with him living with me and keeping the rent that allows him to 'earn' to himself. It is the living arrangement that allows him to rent out his place, so IMO you should get a share of the rent.
His mum is a different question. If she is very old and frail, he might want to see her every day in case it is her last. We are in that situation just now with MIL, and I don't begrudge my husband spending time with her. I go if I can, as I love her too.
It has meant no holiday this year, and limited what else we can do - no full days out etc - but sadly it won't be for long, so it's not a problem.
If, OTOH, she is relatively young, and he is going there for hours a day so she is getting his company and you aren't, it's different. I'd be inclined to suggest he visits on alternate days, but in the end it's up to him. I can just imagine the outcry on here if a DIL stopped sons visiting their mothers. If you don't like it (and I wouldn't in those circumstances) it's maybe time for an ultimatum?
But - it's easy to sit here thinking 'I would do XYZ' without needing to do it - this is your partner, and you need to decide how you'd feel if the relationship ended. What else are you getting out of it, and are you willing to forego those things?