Two pairs against shove

Well I'd bet flop nevertheless. People are expecting to get stabbed so much they call a lot.
Turn he's representing QT so much it should be a fold.

If he has low flop cbet, its reason to stab otf to get value. Just go for value. As played its tough, he could easily shove aa there, maybe even ak. Theoretically definitely call. Explo I might even fold, but pretty tough even though i expect qt this line screams aa to me.

The flop check is not a bad move imo. He was the agressor and you have position. It is not very dangerous indeed.
Check is good because you keep his bluffs and may get value on later streets.
The turn is not the best. Raise him is fine, but when he 3bets shove is a nasty spot. Im not sure what to do here.

I think the flop check is bad. Regardless of villain's stats, I would just bet here. You have top 2 pairs and can probably just bet bet bet if the runout/action is not horrible for you on later streets.

Mandatory flop bet. What do you bet in these spots if you dont bet top two? Bluffs only ?

You also need to bet do deny his equity. For example: He would fold 99 on the flop and you would take the money there, however, he is not putting a single dime unless his improves. If he improves though this means you gave him a free card and now you lose a whole stack. 16% cbet means this guy doesnt cbet his draws so he also has QT in his range here. I would puke fold the turn. You only beat a stupid bluff or J9/K9 which i am confident he would not re-pop on the turn with.