OTF I'd be raising for no other reason than the flush draw out there. Make it unprofitable for him to draw to that. As others have said, if he shoved I'd be calling and if he has hit the set lucky him.

Personally I like the bet ott. It represents the A, makes it look like you've floated his donk bet and hit lucky ott and doesn't give him a free card to draw to a flush or straight. His c/c has me thinking he has a draw or possibly some Tx combo.

As for the river, I'm fine with checking but from what little we know of his play I'm putting him on some non A Tx single pair combo like JT or KT, possibly with a busted FD backing it up or possibly JJ-QQ(unlikely though). I'm thinking KK+ and probably AK would have re-raised preflop. I'm thinking with 20% donk bet he probably donks whenever he hits top pair or has an over pair. He slowed down when the A hit and I'm guessing AT would have c/r us ott. So I'd be happy to shove the river as well but that's just me and I don't know if it would be +EV.