We're playing the deuce-seven propbet for only 20c but I'm always really considering they could have 2-7.

villain is a kinda weak player, bluffs without representing a hand, does not think a lot. Plays a lot of hands even real rubbish.

He also acted before his turn. He thought he was UTG and raised to 70c.

Preflop:
Hero got AQ and is SB
Everybody folds to Hero, Hero raises to 70c, Villain (BB) calls 50c.
Flop: (1.4$)
AQ5
Hero checks, Villain bets 90c, Hero thinks for second and calls.
Turn: (3.6$)
5
Hero checks, Villain bets 1.9$, Hero calls.
River: (7.4$)
A
Hero??

I thought about leading out here, but really big like 10$ or AI for 19$.

What does a $0.20 prop bet have to do with this hand? If people are betting 10X that to win it on a street, you've set up a nice game for yourself, but I don't think it is a factor.

You've cost yourself money by playing this so passively. If has KK, he's not going to bet for you and he's folding to any bet you make. Therefore, you should have made your cbet. If he has AK, he wasn't folding. You could have bet more like $1.2 on the flop and $4 on the turn. That's an extra 10BB in your pocket. It is the difference between a winning player and a BE player.

The river bails you out, so you can jam with ease, because he's calling if he has an ace.

He wanted to raise but he raised out of turn, so my raise from the SB seems very strong right? So I thought I would get more information in checking, because when he would have like a medium pocket pair he might not even wanting to bet with on the flop. So on the flop I thought: he has Ax or 2-7.

I know the 2-7 bounty isn't that much but we always play 2-7 such because it's such a funny game and really good for the dynamics of the table.

I absolutely hate the 7-2 game. Nice for your EV since people will dumbly bet 40x the amount they could possibly win in the prop bet, but it turns the game into an unenjoyable, comical circus.

A new 7-2 game should be created that includes an intelligent player betting a group of morons that there are starting hands worse than 7-2, and collecting from each idiot who argues that there aren't.

As far as the hand, you can't change your betting lines simply because someone might have 7-2, because you'd be playing differently every single hand. Try and build the pot by leading out on the flop or turn. River is either an open-shove or a check depending on your read. Open shoving stacks Ax, speeds up losing to 55, and folds out everything else. Checking gets a check behind from everything other than Ax, 55, and bluffs.

Noticing his betsizes are weak, what would you do?

Flop betsize seems fine, turn is a little small, but not terribly. I'd open shove.

People should get very wary when you check as the preflop aggressor and yet dont give up. Your goal should be to get the stacks in, not milk him on some bluffs imo. Better to cbet and hope he hit the flop in some fashion. Bet flop, bet or checkraise turn if you think he will fire alot and then see if you can get it in by the river.