O prefer bet /fold small OTR:

He would need to turn TP into bluff sometimes for the ch/call to be good, which he probably won't.. this sizing seems to be extremely As heavy.
I like blockbet /folding as well... they won't fold worse flushes, and some weaker guys might even call with worse.

I like to check-call with KsK on this flop. I'd also check nut flush draws and flushes some of the time along with a bunch of AJ KJ and QJ QQ-AA, and underpair combos. I am also check-folding a lot of air on this texture so I feel this is the best way to prevent villain from being able to profitably bet any two against my checking range.

As played I'd check the turn with intention of check-calling. I'd bet hands such as AsKx and AxKs AsTx KsQx KxQs. I think those hands aren't good enough to x/c but have enough equity to be bet. I also x/c turn with AsJ. I'd check all nut flushes.

OTR after your flop and turn lines I think you should bet-fold small-ish (25-50% pot) expecting to get called by some worse hands in villain's range but I don't think he'll ever bluff raise you.
But as played I would just check-fold. I believe that check-calling the river in this spot is the worst line you can take aside from check-raising.

xc vs regs, b/f as a default at these stakes vs unknowns

vs this sizing you can prob overfold even vs regs