I open the cutoff with QQ22x and get 3! by the SB, and I call.

Flop is KQT with a flush draw. I have no flush cards. SB checks, I bet he calls.

He takes 3, I take 2. I catch a brick and, the board is also a brick. Villain checks. Should we be betting turn?

Villain obviously has AA, and my hand looks a lot like 2 pair. I also rate to have something better than AA for the board. Does he have incentive to call turn/river with naked aces? What hands can I get value from?

I assume you meant you drew 1 - he can easily have a flush draw and/or you are freerolling AAJ, I'd risk it

Yes, I drew one.

Probably just resulting myself/telling a bad beat. Of course Villain had AAJ with a second pair and I got scooped.

To answer your original question naked aces are unlikely to call you unless you bluff a great deal. You are unlikely to get value from most hands as the hands that do call will tend to be ahead of your draw hand not behind. Now AAJxx will call some of the time hoping you are not two pair.

First thing to note: The preflop action is a little weird. He 3-bet check/called on KQT. This may always be exactly AA drawing 3, but, if so, your opponent's game is probably not the strongest--And if it does, I'd strongly discount the idea that villain has the As.

To answer your question:
I bet turn and call a raise. Depending on player you can exploitatively fold river (if villain is straightforward and views you as super nitty), but I'm mostly still calling (non-A) rivers hoping to get half.

You see your own hand as pretty face up, but it could easily be worse than it is when you draw 1 and bet flop (eg 2 smaller pairs [trying to get villain to fold the AA or KK chop]). It's a disaster if you bet/fold to bare aces up on the turn, which can (and should) raise pretty much with impunity if villain is confident you had 2 smaller pair. It's an even bigger disaster if you fold and villain is raising something like AAKJT with the royal draw thinking they have the board locked up and can scare you off a two-pair hand that is drawing dead to the board.

Keep in mind the price, when villain raises the turn, you're going to need to call 2 more big bets to win a pot that will ultimately be 10.25big bets. So, assuming villain always bets river and you can never win the whole pot, you still only need to win half of the pot >39% of the time (meaning you should call down if villain is raising all aces up in hand). And with this villain, he might turn his hand face up again and check some rivers, and it's probably likewise okay to fold on A and spade rivers.

He drew three perfect cards and got a clean run out, nothing to be done.

Through a funny coincidence....playing Drawmaha yesterday, I defend the BB with naked AA vs the same Villain. Drew 2 pair, and made a FH on the board.

Villain was so pissed, he racked up and left.

This seems like a super standard bet on the turn and call down if raised