One of the first few hands in the tournament against players I had never played with. What do you think about this line? What do you think about my line? I bet small in case they are slowplaying a set or KK/AA or have a straight and I'm ready to fold to a raise. But I want to bet because if I check and they bet then I'd have no idea on whether my hand could be good or not.

Poker Stars $20.00+$0.80 No Limit Hold'em Tournament - t10/t20 Blinds - 10 players
The Official 2+2 Hand Converter Powered By DeucesCracked.com

MP1: t1500 M = 50
MP2: t1490 M = 49.67
MP3: t1380 M = 46
CO: t1130 M = 37.67
BTN: t1460 M = 48.67
SB: t1480 M = 49.33
BB: t1940 M = 64.67
UTG: t1570 M = 52.33
Hero (UTG+1): t1500 M = 50
UTG+2: t1550 M = 51.67

Pre Flop: (t30) Hero is UTG+1 with Q Q
1 fold, Hero raises to t60, 4 folds, CO calls t60, BTN calls t60, 1 fold, BB calls t40

Flop: (t250) 3 6 4 (4 players)
BB checks, Hero bets t140, CO calls t140, BTN calls t140, BB folds

Turn: (t670) 2 (3 players)
Hero bets t200, CO calls t200, BTN calls t200

River: (t1270) 9 (3 players)
Hero bets t240, CO folds, BTN calls t240

I don't see too much wrong here. You already said they're randoms, and plenty of randoms will be calling every street with AK AQ JJ TT or rag aces which flopped a pair. Your pot control seems good if villain has hit a set and is playing rope-a-dope. I think you're striking a good balance between extracting the maximum when ahead while at the same time losing pretty close to the minimum on the occasions you're behind.

What do you do if BTN shoves river?

ugh, now i remember why I hate DoNs. I think I bet bigger on the flop and then slow down on turn and river. I hate getting a lot of chips involved early with QQ, even if theres not an overcard.

If BTN shoves I think I can only beat a bluff (I don't think he'd shove the river with JJ/TT without raising an earlier street), and I don't think a random would bluff. Also, in that board there were not many possible drawing hands. So I'd fold.