What kind of player it is and what kind of range my opponent is on-- calling a UTG raise from the BB, check/calling a 52T rainbow flop, then checking the turn, is more important to me then the K that hits the turn.

It looks like a sign of weakness, but he could also be pulling a check/call + check/raise line with a set. In any case, I see different arguments for how to play the turn but I think it's more a matter of feel for the opponent. Keep in mind he is letting you see a free river and what looks to be 10 good outs, 4 of which are the nuts. I usually take that here...

On the river, I call and not only because I want to see his cards, (which is obviously a wealth of information) but I'm getting 15:1 here and I think I'm good here more than 1 out of 15 times.