1) m/r on the flop = monster (set or straight)... might be AQ. Probably not a flush draw. Turn is interesting. He does not look like he will fold if you raise (!) but you might have a huge number of outs if he has 2p/set (maybe even 18 outs if he has JT). If he does have a crackable hand (i.e., not AK with Aclubs!) then he bet a little bit small. He might even be scared of the flush and check river. I could see a call or fold here.

2) if you don't want to gamb00l, check behind on the turn. If he as a flush/worse draw than you then he made a HUGE mistake checking, becuase a 4th diamond and you will likely stack him on the river. As played... shove or fold. Or call. They are all kind of equally crappy.

3) I probably just check like a nit here, but I'm not worried about squeezing value out of my KK on a paired/3flush board. You can't bet/fold. There are a few hands you beat that might call a small bet, A9, JJ, 89, AQd. He probably has 2 diamonds It might even be worth checking for info (when he would fold to a bet).