$50 nl, flopped set on broadway board, line check

$50 nl, bb=$.50

Relevant stacks
Villain $103
Hero $57

No reads on villain except that he has doubled up on the maximum buyin and I have not noted any donktastic plays.

Dealt to Hero JcJs

PREFLOP: Villain limps, Hero raises to $2.50, one caller, villain calls.

FLOP ($8 in pot) Kc Jh 10c
Villain checks, hero bets $4, third player folds, villain calls.

TURN ($16 pot) 3h
Villain checks, Hero bets $12, Villain calls

River ($42 in pot) Ad
Villain pushes relatively quickly, Hero pukes, wishes he could find a call and be right, pukes again, and folds.

I have been mainly a Sit and go player and am trying to transition to some more cash game play. I feel pretty good about the preflop raise, but comments appreciated. On the flop, what is your play? I half potted because I wanted it to appear like a continuation bet and hoped top pair or two pair might come over the top. Also wanted to make draws pay but also wanted to wait for turn so I could get away if a terrible card like the 9c or Ac showed up. On the turn, should I have full potted instead of 3/4? I still wanted to extract value and not overbet push. Finally, on river I just couldn't find a call without having actually seen the villain either (a) make a smaller value bet when hitting an obvious straight, or (b) bluffing all in on an obvious scare card. Without this read, I feel like it is unprofitable because too often he is check calling with the Q (and maybe a K or 10 for a pair) and pushing when his straight hits. Absent a specific read, can anyone find a call here?

Comments on any or all streets appreciated.

[Fallen Hero]

You can't half pot that flop, that's an incredibly draw heavy flop, if you're going to pot any flop, that's the one. the turn looks nice, and I fold the river (even though it's a pretty good move if he does it with a busted flush draw)

PSB the flop. For good measure do it again on the turn, if you must, but that's it. You will often get killed on a flop like this and its best to find out early. A call on the flop bet should arouse your suspicions fully. Try to get to the river cheaply is my usual strategy here if he calls flop.
fold the river, he's called big bets with something big . and note he's no longer passive.

Thanks for the input, fellas. I felt I should have bet more on flop. I guess the reason I wished I could have found a call is that I have pulled this bluff out when I missed my flush draw before, so I know there are people that are capable. Like I said, I couldn't pull trigger on a huge call like that without a read to be suspicious of villain in particular.

If I had potted on the flop (or even if I bet the same) and (a) got raised some amount (but not all in) or (b) it was check pushed all in, what do you do with a set on this board in those scenarios?

on the flop? I'm not tight enough to fold a set on that board on the flop. If he pushes the turn over my PSB I can fold.

and the river is an insta muck, but its much easier if you pot the flop and turn