Mostly I'd say you played it fine.
Your stack size and X2 pre flop raise labels you as a beginner. So play with 100bb so that when you do win a hand you get max value out of it.
We have no stats for the opponent, if you have been paying attention you should have some idea of how loose/tight and passive/aggressive he is. Knowing these tenancies is very important to understand what his bets mean.
so he opens from early position and generally this means he has a pair or a decent ace. Once you understand how often someone plays from each position on the table then you can assign an appropriate range of
hands.
Generally donking into you on the flop is just really bad here he has no idea what you have.
You must examine the flop carefully every hand. This flop shouldn't hit most players 3bet calling hands. The best hand would be a set which is possible of course but donking out near full pot with sets is just going to fold out a lot of your hands that you might have cbet with. Generally if he had a set a check raise is the better EV line.
Of course if you are playing very tight and he can safely put you on a premium over pair then donking to build the pot knowing that you aren't folding early in the hand is okay.
So you see it all depends on a lot of factors. Aggressive players like to donk lead the flop with their air just hoping you will fold, passive players rarely do this.
It could be that he has Jx here hands like AJ KJ QJ TJ are all possible and probably more likely than sets. But leading out on the flop with any jack hand for value is still pretty dense bearing in mind you have a fair amount of over pairs.
There shouldn't be any 2 pair hands other than an occasional 76s. So now we consider the draws.
89s and JTs are possible but unlikely as most player would fold these hands out of position to a 3bet.
So I'd put him on either just over cards, Jx a set, or 89s in that order of likelyhood.
So the turn is a Q and he check calls. Your turn sizing is pot committing ie too large as really now you have so much money in the pot that you shouldn't be folding hardly ever on the river.
The upside is the quality of the information you get when he calls a turn bet of this size. Now he's telling you he has a good hand maybe one that now beats you. JQ becomes a hand to fear and the sets now all become more likely. He may also have KQ and AQ now if he led out with just overcards.
I'd have bet around 35-40c max on the turn. Enough to charge his draws and keep in his weaker JX and charge his Qx
A good player who can beat AA here would have just put you all in on the turn here knowing that you are pot committed.
The fact that he doesn't raise the turn signifies a little weakness and as this is 2nl I still wouldn't rule out 89s and maybe JTs at a push.
When he checks the river I'd say he has AJ KJ QJ or Qx and still some sets if he is a passive type.
You may as well shove in the rest on the end as he's getting good
odds to call and he should call all his Qx. You will be beaten some of the time the trick is not to pot commit yourself on the turn. This is another failure of the stack size you started the hand with.
If you had 100bb it's call flop, just over half pot turn and river folding to any turn or river raise.