1st hand -- I say re-raise the flop to $5, this will let him know you have better than TPTK. This also should give you control of the hand since you have position. If he re-raises you are beat, if he flat calls you are not in great shape but you have control.
I don't see him having 64, 58 or 53 with your pre-flop raise so the straight is out. I don't fear a set here: KK is possible but not likely since no re-reraise pre-flop, 66 and 44 are possible, but I pay them off here. I don't see him calling a 6x PFR with K6 or K4. I think you laid down the best hand to a TPTK.

2nd hand -- I think you know you shouldn't have pushed with flush on the board.