Well, I think the small 3bet in the first hand of the match can mean one of two things: either he's an aggro donk and does this with close to any two cards, or he really has a hand and is either trapping or doesn't know how to 3bet properly.
In either case, I don't think Q5s plays well enough post-flop to warrant a call.  Sure, you're getting 4:1 in position, but you have no idea where you stand post-flop and have no reads whatsoever about this particular opponent.  I wouldn't mind calling here with a hand like K8s (where I have a chance of flopping top pair with both my holecards) or a suited connector - just something which plays well post-flop.
IMHO folding here pre-flop isn't like giving up on a chance which won't come back easily - if he's an aggro donkey, he'll surely do more spewy plays later in the match, when I have a stronger hand and/or a better idea of his tendencies - and against someone who's doing this with a real hand, my Queen will never be good post-flop and you basically play this in the hopes of moving off his hand somehow.  Another thing to consider is that it's not just the 30 you need to call here - you'll almost always get a continuation bet at the flop without knowing anything about this opponent.
Things would be a lot different if he already did these small 3bets - or other spewy things - a few times before, so I could reasonably assume that I'm not up against some misplayed Aces or KQs or something like that.
Post flop - call me weak - but I'd even consider folding this on the flop.
The way I see this is, if you want to continue with that hand, then it's either going to showdown, or you'll fold at a later street after putting a lot more money into the put.  I don't think a guy who min3bets you like this in the first hand will ever bet this board then give up, checking it down, I'd almost always expect to see a turn bet from him if I choose to flat-call this flop.  So there's basically no way of seeing a cheap showdown and - to make things worse - due to the many draws, you have no idea which turn and/or river cards help your opponent and which don't, so there's basically no "safe turn" or "safe river" for you.  Also, if you flat-call a flop like this, you pretty much turn your hand face-up - you either have a weak made hand which can't stand much heat or a draw.
Because of this, I think the only way of continuing with the hand on a board like this is to get the chips in on the flop - but I don't think this'd ever be profitable without knowing anything about this opponent.  Call me weak, but if I have the choice between committing all my chips or giving up - and I have no idea whatsoever about my opponent's ranges and tendencies - then I always decide to fold if I haven't already invested a lot of chips into the hand and I also have enough chips to continue and wait for a better spot.
I'd simply give him credit for a hand post-flop and see whether he does these small 3bets more often - and if I do call the small 3bet, then I'd fold on the flop, I simply don't want to call two more bets without having an idea where I stand.  If I played him before and knew he was a huge donkey who's spewing off a lot of chips with a draw, then I'd be much more inclined to continue with the hand, but not without any reads at all.
Jack

I call min-3bets preflop with almost ATC if I'm deep like that. So I think preflop you played it fine.
However, the flop is so incredibly wet, I don't want to see another card...and your kicker sucks. Unfortunately you don't have any reads, so it's not easy to choose the right action.
On one hand, you have TP, on the other, the guy 3bet you preflop which generally indicates 2 broadways at the 11s. I think I just often fold the flop without reads because most cards which could come on the turn/river will sting like hell. The thing that really irks me is his river shove. If he had a strong hand, I just don't see him shoving after betting so weakly for 2 streets. I can't see the 7 or 6 helping his hand at all. You don't see too many players 3bet 66/77...
If we give him a 3bet range of TT+, AJ+, and KQ...which seems to be the general 3bet range at the 11s, then you are a 59% favorite on the river. Given that he's bet weakly every street but the flop, and the turn/river cards haven't really changed all that much, I think I would call this. I just don't see too many ppl 3bet with hands like KJ, or J8 at that level...so the only thing I'd be worried about is KK, or trips.
Cliff notes: Poker Stove says it's a call against a random player's likely 3bet range, we're a 59% favorite. I would sometimes fold this on the flop, because most turn/river cards will hurt us, but as played, I call the river. His river shove is so weird though, that I put him on the nuts or air (AK).

Hmm, you're probably right about the river if we can assume that he's somewhat reasonable - but I'd still fold this on the flop because I'd really expect to face a higher turn bet from him, something around 2/3 pot size, and there are als o too many turn cards we don't want to see.
However, I'm not so sure whether we're still a favorite at the river if the guy is some weird donkey - I've seen a lot of bad players at the lower limits do these min3bets with a very wide range, especially early in the match, and I've also seen a lot of them take these weird line of betting the turn so small and then shoving the river if they hit the nuts on the turn and want us to shove over them.
If I see this correctly, our equity at the river also goes down if we put more Queens into his range - but doesn't your 59% mean that he'd be shoving the river with his entire 3betting range ?  Not so sure whether a tight player actually does that on a board like that.
However, I agree that his shove is either the nuts or air.
But doesn't this mean we're much more likely ahead if he's a tight player than if he's a complete donkey.  I mean, if his range is so polarized, then we can probably remove all better queens from his range and can basically give him a range of Aces, Kings, Set of Queens (I know the small turnbet doesn't make any sense with these hands, but many bad players on the lower limits either don't know or don't care about protection), or Air if he's tight.  If he's a donkey, however, he can have all these hands in this 3betting range which hit two pair or straights on this board.