On a lot of turns I would bet but on this one I would continue to check. With the 9 pairing you no longer get value from a 9x and you're behind a Kx so by betting for value you're trying to get calls from a 6x or mid pocket pair. You get value from draws as well but I typically expect players to bluffs those when checked to so you're likely not missing much value there.

If you bet turn and get called then river is probably a x/f considering the flush got there and you're still behind any Kx or 9x.

Yep seems like a legit bet on turn and river and I would go around 50% of pot on both of those streets.

I like a check as'well and get 1 street of value on the river in these spots. I think you can check a large % on the turn even if it doesn't pair ,your really only concerned about an Ax so 3 outs i don't think your hand needs much protection. If you make a small river bet around 1/3p youll just get snapped of so wide . I think making bigger river bets just gets to many folds when their ranges are so weak, I don't really see 2 streets of value in these spots maybe i need to adjust my thinking a little.

I would have bet the flop anyways.