I believe the question asked was: what is our calling range. I don't think he meant what we should do when we have air or when we have the nuts, but what we should do with a marginal hand. I.E. what hands do we consider bluff catchers here?

I asked my previous question because maybe we should consider a different action on the turn given our hand other than bet/folding (because if we're bet/calling an all in on the turn I don't see why we would ever check/fold this river).

This is a really bad spot to bluff for villain and he might not even try and valuebet AQ here, depending on how solid he is and what he thinks of you.

However, with your read:

I stole a pot from him lately by c/raising as he seemed to bet when checked to.

I'd probably fold AJ as played and call AQ/AK.

With that same read I dont see why you would ever bet that Ace on the turn if you actually have one (not much value in it other than AK), especially since he bets like a monkey when checked to. I'd check that turn with all aces and just expect to get to showdown and expect to see a bluff here sometimes, and sets the other times.