“Looking at the title, you pros will just shut your browser tabs saying “This is an easy fold. Why is this bugger getting into dicey situations with this hand?””

That’s not fair, I love getting into dicey spots with marginal hands!

“I raise to take down blinds and antes”

That’s not a thing you can do. I mean, you can’t just say, “I’d like to take down this pot,” and then do it. The value of any pre-flop raise you make is going to be some combination of the fold equity (what you win when no one calls) and its expected value when called or three-bet. I’m not taking issue with your raise, I think it’s a clear open with this hand from this position, in no small part because the hand will play well when called and can even peal a small 3b like this one. I would, however, make a smaller open. You may not want to raise the minimum, but I think anything over 1000 is sub-optimal unless the table is super-loose.

“I check-called to represent the A so that he will shut down betting in future streets.”

Maybe he will, maybe he won’t. It’s OK/correct for you to have a range that calls once and folds to further action unless it improves, and given the price you’re getting, this is a fine hand to put in that range even if Villain will have a bluffing range on the next street. I wouldn’t say that you’re calling as a bluff, and I wouldn’t recommend getting aggressive unimproved in attempt to bluff him off of KK or anything. You have a shred of showdown value, you have a chance of improving, and you’re getting a good price to call. It’s fine.

“If I had a big ace, I would have 4-bet pre.”

Really? With AQ/AJ/AT? I wouldn’t recommend it.

Why do you think you should have folded this hand pre-flop?

I’m not sure if you mean to ask about the turn, but it’s an easy check-call precisely because Villain can have AK. I don’t see why he can’t have two pair, but just because he could doesn’t make your call bad.

PS What is “a good image”?

If you really think V’s range is that tight, I guess it’s a flop fold, but I’d question how certain you are that it’s that tight, and also whether V would bet hands like KK or QQ (or AA, for that matter).

“Might even be drawing dead” is already factored into the price you’re getting. You don’t need to consider it as a separate argument for folding.

What’s the value in 4betting AQ if you think V’s 3bet range is that tight?

As for calling the 3bet, you have to pay attention to the price you’re getting (which you did here). You can’t just create categories of hands that call 3bets and hands that fold to 3bets – it has to depend on the size of the 3bet, the 3better’s likely range, etc. But with the right price and/or if his range is wide enough, excluding some monsters from his range is not a prerequisite.

Why is it good that people aren’t jamming QJ over your raise?