Can I get away from this hand?

How would you play it? I bet very little since I wasn't afraid of any draws, just overs, or maybe other pairs hitting a FH.

Read: BB is pretty retarded, saw him calling AI with JTs against AKs.

PokerStars No-Limit Hold'em Tourney, Big Blind is t200 (8 handed) from (Format: FlopTurnRiver)

MP2 (t2440)
CO (t4920)
Button (t1605)
SB (t6140)
BB (t14380)
Hero (t6178)
UTG+1 (t3150)
MP1 (t3805)

Preflop: Hero is UTG with Q, Q.
Hero raises to t800, 2 folds, MP2 calls t800, 3 folds, BB calls t600.

Flop: (t2500) 7, 7, 6 (3 players)
BB checks, Hero bets t1000, MP2 raises to t1640, BB calls t1640, Hero calls t640.

Turn: (t7420) 9 (3 players)
BB checks, Hero bets t3738 (All-In), BB calls t3738.

River: (t14896) J (3 players, 1 all-in)

Final Pot: t14896

Results in white below:
BB has 8h 5h (straight, nine high).
Hero has Qs Qc (two pair, queens and sevens).
MP2 has 5c 5s (two pair, sevens and fives).
Outcome: BB wins t14896.

Every draw hit and the board is paired...I think it's fairly easy to throw away 2 queens there.

Hi Dafish - how ya been?

Simple answer to your question "Can I get away from this hand?" is probbly yes - on the turn.
Following your flop leadout of 1000 MP2 calls all-in so you have to figure a 7 or spades or possibly another pocket pair but then BB also calls - so what does have thats worth t1640?
Here you had a chance to isolate MP2 with an all-in that should force BB to fold flush/str8 draws but after seeing the results this donk might have figured he could afford an 8-out draw.
Turn comes up a spade and that's where I would have checked it through. You still only have one overpair to a paired board and both a str8 and flush draw staring you in the face. Check and fold on the river when BB comes out for the rest of your stack!

How did every draw hit here? We can ignore the short stack, he could have a wide range of hands and probably did not hit. He could be bluffing or have a very weak draw. Given the pot size he can pushing any 2 cards here. The only problem is BB. Sure he could have a FD, but that still doesn't make him a favorite to win the hand.

Just make a 2000 chip bet on the flop and it plays itself.