Situation: A rather passive 6-max Cashgame, Villain is on the more aggressive side of things, compared to the rest of the table.

Poker Stars $0.01/$0.02 Pot Limit Omaha Hi/Lo - 4 players - View hand 3080679
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BB: $1.61
CO: $7.42
BTN: $2.08
Hero (SB): $2.20

Pre Flop: ($0.03) Hero is SB with A Q T 2
2 folds, Hero raises to $0.06, BB calls $0.04
Villain usually raises good hands, so I have no reason to believe, I'm not really strong here and raise it up.

Flop: ($0.12) 5 2 Q (2 players)
Hero checks, BB bets $0.02, Hero calls $0.02
Okay: Akward flop. How do you play this spot in general?
I check, because I don't expect to get called by worse here ever. However, I'm not folding to this tiny bet.
* Are there reasons to bet?
* Should I fold, if he had bet more?
* Are there reasons to raise?

Turn: ($0.16) K (2 players)
Hero checks, BB bets $0.04, Hero calls $0.04
I like and I don't like the turn. I like it, because it's a high card, so I'm not getting freerolled by some made low, I don't like it, because it is an overcard to my queen.

Another tiny bet, another moment, where I can't find a fold. Is this wrong?

River: ($0.24) Q (2 players)
Hero checks, BB bets $0.10, Hero calls $0.10
1) I considered making a huge bet for a moment. Obviously QK and KK are always calling, but maybe I can make a Q5 fold.
I decided against, because I wouldn't have played QK/KK this way and it's a 1c/2c table, people will never fold any full house ever.
2) I decide to check/call and add myself to the list of people that can't fold the underfull. Is this a clear mistake? (Assuming the villain doesn't really bluff that often, if ever, but might sometimes be valuebetting worse)

He shows Ad Js Qh Ks and wins.

I don't see anything wrong with your play, but it's good that you included your thoughts, because I think there are many things you need to re-think.

You are way way too scared of running into a huge hand in a HU pot.

Check-calling the flop is not bad per se I think, but there is definitely nothing wrong with betting. As played, turn is a check-call. River is a fist pump call, honestly I can think of many players against whom I'd consider raising for value. Folding is just plain bad, unless his perception of ranges is as skewed as yours is. In his shoes I'd be value betting any Q.

Betting the river doesn't make much sense if you think about the previous streets, but if you bet, you bet for value and not to make Q5 fold.

I'm not sure it's even accurate to describe your hand as an under full house in this situation.

On the river, you lose to these full houses:
KK (3 combos)
QK (3 combos)
Q5 (3 combos)

You tie this full house:
Q2 (2 combos)

You beat these full houses:
55 (3 combos)
22 (1 combo)

So even if you know for certain that Villain has a full house, he only beats yours 9 times out of 15. But you don't know that for certain. As amok said, he could be value-betting AQ (or even AK or AA, given the weakness you've shown on all three streets post-flop). He could also be bluffing with a missed draw.

Look at all the draws on the turn: low, wheel, flush, Broadway. The pair makes them all miss. This is a prime bluffing opportunity if Villain missed all his draws and hopes you did too; it's therefore a prime bluff-catching opportunity. And you have way more than just a bluff catcher here.

An under full house is when I have pocket deuces and the flop comes 244. Yuck. Now when a 6 comes on the turn and two other players get into a raising war, is it because one has A3 and the other has 35, or does someone have 46 or 66 for a bigger boat than mine? That's the type of situation you want to get away from.

Thanks to both of you for your feedback. I appreciate it.
I am wondering, what I can do.

I started playing lots of headsup sit & gos, but it is just not the same thing. (I do pretty well there and don't play nearly as weak and passive as here.
However, often times, when I try to play the same way in a cashgame, I keep running into those big hands and losing lots of money in the process.

On the other hand, I know from all the tourneys I played. My postflop play is bad. Often times, I make my chips by doubling up through people that call me with way weaker hands and then losing them slowly, whenever I play a proper pot.

About the cashgames in particular. There is pretty much three types of players:
1) Guy who raises every hand and pots every street until you finally fold.
2) Guy who never raises, only plays very few hands. When he calls, he usually has the nuts in one direction and he never bluffs.
3) All the people new to the game. Usually playing every hand, check/calling every street until they get stacked by 1) or 2).

Here I think, my villain is mostly a type 2) player and I think, he's would only bluff very rarerly, if ever.