Call 3bet Pre, bet the Turn?

I would fold to his 3bet preflop.
But I like the rest of the hand. Bet OTT seems reasonable after he checks, plus you have a gutshot there. OTR I would check behind as you did.

Preflop it depends on your OR from button, in my case KTo is bottom my 3bet calling range, but it depends on sizing too.
OTT I don't like this bet, because we have quite good showdown value, we don't need to protect our hand, bc his straight draws he should bets on the turn as well. On the turn like this, he can have some Jx hands, overpairs which beat us and he checks because of pot control. And he has some give up bluffs, but he would go check/fold, so we don't get any value from these hands. So I don't see any reason for bet ott. Maybe on the blank river if he would check, sometimes we can go for thin value (if you check back ott of course).

Preflop: At higher stakes this is an easy call.
At lower stakes if the opponent is only 3betting 8-10% you might be able to find a fold with KTo. Vs a 13%+ range KTo is a defend. Consider using KTo as a 4bet bluff if it's too weak to call 3bets since you have blockers to AK/TT/KK.

Turn: Take your cheap showdown.
a) On this coordinated turn it's the right play to check/call a lot with TP/Overpairs. And if you want to bet turn (which you shouldn't because this is a medium strength hand with midpair), then you need to bluff river as well.
b) You are good here a lot. Betting here is only good if you expect him to bet Flop check/call turn with many worse hands (which I don't think he does, because stuff like AT KT QT will often check the flop)