A few observations:
1. I think it is important to remember, OP, that you say that villain "calls way too much before the flop," despite your impression of him as a winning player. If this is so, and the BB is a donk, then you absolutely should have 3! preflop because failing to do so: a) got you into the predicaments you faced on all subsequent streets (i.e., not knowing where you were at in the hand); and b) you likely missed value preflop with the best of it.
At the heart of your inquiries is this problem: by not taking the lead pre and then again on the flop, you put yourself in a position where you did not know whether you were trying to extract value from your hand, or whether your hand was second or third best and had essentially been rendered a broadway draw with very marginal showdown value (unimproved) in a 3-way pot. One of your primary objectives in betting out the flop should have been to try to induce a fold from the BB, even if he is a loose player.
Had you made it 3 bets to see the flop, you'd have taken the lead in the hand, and the flop should then be an insta-bet. Had you been raised by villain, that would have been OK in the sense that you would have at least acquired some sense
of where the BB was at in this hand, based on his action subsequent to the villain's raise. As played, you had no idea whether your hand was best, 2nd best, or 3rd best! BB could have held T-rag, Q-rag, a random underpair, or, just as likely, a hand like K-T or even K-J, for bottom two pair. You just had no idea! But had you bet out and villain had raised, you should at that point have suspected that you were behind one
of the two hands you were up against (in the event that the BB had called two bets on the flop), in which case your hand was pretty much rendered a drawing hand, to be played differently than you'd play A-Q to extract maximum value with top pair. Ceding control of the hand to the villain by checking this flop cost you a great deal of information--not only about his potential holding, but also the BB's.
2. You say that had the BB called two bets on the flop, you would have--at that point--raised for value. This does not make sense to me. If you knew you were getting action from villain, you should have lead out and if the BB had folded to villain's call or raise, that would have been GREAT! You did not want the BB calling down with hands that could make two pair or a straight here. If BB called two bets cold on this flop, it was not at all clear that any subsequent 3! from you would have been a good raise "for value." The hands which you were hoping
the BB held--had he called 2 bets cold on this flop--were exactly: A-Q (your holding); A-T; K-Q; K-10; Q-J; and J-10. The other possible broadway combo's which he could have had here were: A-K (not likely, as played); K-J (bottom two); and Q-T (the nuts)--all hands which beat you. So if you were contemplating raising the BB "for value," he would have had to hold exactly one of the first 4 combo's for your play to be correct. (Note that I am assuming he would have dumped hands like Q-rag suited and K-rag suited facing a third bet... if this assumption is in error, then the analysis changes somewhat...). But the larger point is that this is not a spot to induce value-calls from the BB, but rather try to induce his fold!
By taking a passive line on the flop (not betting out), you may have induced the BB to chase you in this hand where he otherwise would have folded because he likely suspected he had the second best hand and was really only competing for the pot with villain. Given his range of possible holdings (as above), this was not a good thing! Your hand's equity in this pot decreased dramatically if the BB was holding a pair and a straight draw, especially if your kicker card would have made the BB's broadway.
3. Had you 3! pre, bet the flop, and called any raise on the flop from villain, I believe your line on the turn would have been much more self-evident, especially if this had gotten the pot heads up between you and him. As you played this hand, both the turn and the river were going to be tricky no matter what came. Again, we had no idea if A-Q was good on this board, given its texture, even if the turn and
the river had bricked off!
What card would we MOST like to have seen on the river? If we hit one of the 3 remaining sixes (to make two pair), any A-rag played by either opponent was chopping half the pot with us (the K on the board plays). Gross. If ANY broadway card besides a T had come, we were likely screwed. (However, had we gotten the pot heads-up with villain, we know we ABSOLUTELY would have bet out to induce a fold--or at least to find out where we were at in the hand--in the event of any non-T broadway turn card. Had a T come, it would have been an obv ch/r given the villain's tendencies and the price the pot was laying him at this point in the hand.)
The best case scenarios, as played, were for a 9, 8, 7, 5, 4, 3, or 2 to come on the river, increasing the likelihood that our hand would win unimproved in a showdown where we had no choice but to call down with our A-good kicker.
So another big part of the problem with the passive way you played this hand was that you made it harder to semi-bluff a lot of turn cards (particularly any broadway cards)--cards which, by firing into the pot heads up against villain, very well could have induced his fold. Note, too, that while these same broadway cards would have presented a good opportunity for a semi-bluff-turn-bet HU (if such cards did not make villain's hand, they were good scare cards for your semi-bluff), these very same cards were terrible
cards for you to call down with--especially with two
opponents in the hand--as they greatly increased the likelihood that one of your opponents made two pair or a straight.
4. As played, call and pray.
Thanks for posting this hand. It really made me think through--precisely
--how I handle this relatively common scenario (when you make top pair with your A but the board is all-broadway cards).