H1: I don't mind your line of thinking here at all, and there's really a lot of ways to play this hand. Perhaps if you had more reads then we could make a more optimal decision but if your goal was to represent weakness then checking the turn isn't bad. You could always of course to bet turn and hope he thinks you're bluffing which isn't bad here against a thinking player because it's harder for him to believe you got an ace considering there are 2 aces on the board, but I mean this decision is marginal imo.

The river shove does kind of concern me here though and I don't really think he would ever be bluffing here, at least not by shoving, if he was bluffing I think he'd prolly not bet more than pot size... Anyways, our hand is too strong too fold, the only real hand in his range I think he could possibly have that beats us is a backdoor flush draw which is very unfortunate. I'd prolly make a crying call and hope he doesn't have a backdoor flush draw of some type (AKs, AQs) lol. I'd actually be quite surprised to see him with 77 as I'm not sure he'd call with that hand preflop to 3bet just to try and hit a set.

Would be interested in result of H1

H2. I think this is badly played on your part. The check/raise itself is meh but I don't really see what you're folding out but cbets with air and this board hits so many hands that it's not a good board to cbet nor to c/r without something at least. In this case you got a straight draw but even that isn't very strong in this situation, and shoving over his 3bet is a really bad decision imo as you don't have much fold equity and the worst hand he shows up with here is a TPTK/TPGK type of hand, or a better straight draw than yours (although this is extremely unlikely given that he is a nit and raised i EP). Another thing to consider is that he may very well be holding a pair of your outs.

I plugged in some numbers in Pokerstove of villain's range here on what I think may 3bet you:
You got 97s and his rather reasonably 3bet range imo for a nit is 88+, ATs and 44. Basically your only outs are a 6 and a J to make the straight in which you got 1/3 equity. Check/call, or c/r fold, or donk fold here imo is much better than shoving as a bluff.

H3. Pretty easy call here