When i first looked at the hand i would have called flop bet but now after thinking about it i would say that fold on the flop is the best line even tough it seems very tight. Once HJ cbets this flop into 3 ppl his range must be really strong. I would say sets, NFD with maybe a pair or over pair.

As you mentioned raising is not an option cuz we will never get it in good here in a mutliway pot. Calling is also very tough because it is multiway and you still have wto players behind you.

Your equity really goes down if it gets 3 way because often someone with a high flush draw will come a long and maybe some one will have 2pair or better in this spot. You only have 2 outs for the nuts on the turn and a GS which would give u 2nd nuts but sometimes will be dominated by a hand like 678x that would call a flop bet!

This is why i think folding the flop is often the best option...

These QQ are good enough for me to 3b PF.  I am double suited and I want to clean up my flush draws, never want to see too many hands four or five way pots.  We are inviting SB and BB into the hand are going to only feel good if we flop top set.  FWIW I could be convinced not to 3b with the 37 side cards but think being ds is a huge benefit here.

With a 3b we can increase the comfort of our FD and change the stack off dynamics on the flop, while being in position.  We could have a villain put us on AA, KK and stack with QT hand that we have in jail. 

As played a fold is fine for all the reasons given.

Standard flat preflop and prudent flop fold in a MW pot with two players left to act.

I would lean towards 3betting vs a HIJ range we can assume is looser than average with at least some degree of certainty and a weak CO flatting range. 

As played your suits are often 5 way and our made hand is often dead.  Folding seems best.

If folding is correct, I know why uncapped PLO will never be popular. Games that reward nittery are terrible games.