Hmm the guy is repping badly played nuts/missed draws by checking the turn back and then potting the river.

I'd be calling and taking note in this spot, GG if they checked back a full house on the turn.

I like a c-c on the flop sometimes, we have a pair+oesd, I'm not loving life if we bet and they raise us.

Flop you can go either way. Turn is good.

River is a tough spot. I agree with @Harvie that if this is the nuts he played it very weird. Remember, he called flop so it's not like he shows up with a random 65s here. I can think of several value hands he might play this way (I don't think its the best way, but I see how he might). I struggle to find too many bluffs unless he's calling preflop with a bunch of T8 T7 T6 type stuff. Or maybe he called flop with an Ax gutshot? Then again, those are real possibilities.

I'm a little torn here so I guess I think it's close. I'd be ok with either a call or a fold but I wouldn't be thrilled about either option. Worst answer ever, sorry!