Pre flop normally i fold this hand in this position but villain were 3 betting a lot ( even suited gappers) so i thought this hand playing well vs wide range in position. Prob fold is better vs this sizing.

On the flop viallains connect this board quite well so not much reason to betting, maybe small like 1/3 to fold any missed hand and not giving free equity but check seems reasonable as well.

When villain checking this turn he's most of the time capped to weak one pair hand where i can have most of boats like QQ( if not 4 bet pre) not sure if checking weak Tx like QT or trapping TT,KT but can have straight as well. Wanted to bet turn and polorarize on the river, overbetting. But wasn't sure if isn't better to start polorazing from turn betting bigger or even giving up? But when he checking hes capped most of the time

He should't have much Kx but can check call turn with AK,KJ. I have only 4 high but got confussed on the river, prob should bet but what size? Will he call with Qx? Betting small or overbetting saying i have K or nothing?