Since you are OOP I agree with you, that you should make your sizing little bit bigger.
In my opinion the hand that makes the most sense from his perspective is Axhh. He might try to blow you off your equity on the flop, and when he hits his A he might try to pot control. Most likely he doesn't have sets, since he checked on the turn, there is no straight combos that would make sense here. He might have occasional 98 on the river, but I doubt that instead of betting on turn, where scare card comes, he would decide to bet on the river. So I would fold on the river.

Preflop i would def reraise, to about 1,5 dollars, for value
You could either check( with intension of cr or bet on the flop) or bet. Very wet board, so if i was to bet, i would bet big like 3/4 or pot.
Check on the turn is good
Would call on the river. He will barrel his aces for sure, so never think he has that. He would prob also barrel his 5 though there is some merit, in checking, since he does not want to be cr of his hand
the only fives he has in his range is pocket fives, so therefore it is very unlikely he has a five
Also there is a busted heart draw out there, which he also could have

pre flop i agree raise bigger
on the flop i like a check on this bored but after betting you need to call,
on the turn i dont see how you can say he has no draws by checking "he would bet all his draws" if he had a made hand like set or two pair he would bet so when he checks i think its more of a chance he could have draw too thinking your not gonna fold an Ax the way he played the hand it seems like an Axhh type hand but checking turn and betting river i believe your gonna be behind most of the time there