zillagio

Hmm....I would think calling the 3bet is profitable here? Obviously, it's not a great spot, but I think we keep our range uncapped when we call the 3bet so we might not get pushed off on a lot of run outs, we could make our hand on turn, villain could be raising with lower flushes/multi-draws, etc. For the price and stack sizes, I think I call.
Also, I think we can have no check raising range in this spot and not give away too much.

I would fold turn, probably flop, especially given stack sizes and lack of backdoor equities for our hand. I think we should give significant weight to his UTG open and 4bet, not to mention flop cbet and turn shove. There just aren't many combos of airs or draws that we beat, especially when we block a lot of his AQ and QJ combos. Another problem with calling off with our hand here is that now he knows that our calling range is fairly wide in this kind of spot so he will be able to extract more value with his strong hands; that is, by making his bluffs very unprofitable (which he now knows), we make his value hands hugely profitable.

As played, call turn. You say you hated this spot, but it's actually pretty great IMO. We opened in the BB and flopped a set against weak opponents, we have the best hand fairly often, we have enough outs vs flushes, we are in position, and there's still more than pot size bet left on river after the turn call. Obviously, there are better turn cards for us than the Qh but it shouldn't prevent us from playing this hand profitably.
I would raise a little bigger pre (or check) and bet less than pot on flop.