[1] In the CO, I'd see us open with a rather wide range, especially when the button is tight and not difficult to play against. So I'd say the range would be something like 22+, A5s+, ATo+, KTo+, any suited connector, any connected cards from JTo+.

[2] This is not a very good flop for a c-Bet because of the lack of high cards. But if the button plays fit or fold, I'd assume that he has a high fold-to-cbet, so I'd see us betting the whole range here.

[3] The button has called our c-bet, so obviously his hand has sufficient equity against the T+ (more likely) / overpair (less likely) we're representing, so I guess his range includes a straight draw or a T with good kicker.

Since this the flop didn't have a super drawy board, the straight draw should less than 50% of his range. So I guess it's questionable to 2-barrel with total air here. We might be pricing our river card for our 34/78/89/QK with a straight draw or two spades with a decent high card strength for a flush draw. Or we're betting for value with AT / JT / QQ+ or a set.

[4] So we're betting again. This generally means that either we have no showdown value (like a missed draw) and we are hoping for our opponent to fold a better hand, or we have a very strong hand that might get called by a strong, but weaker hand. Since the board is paired, we could only really expect an Tx type of hand to call, so the non-air part of our range would be 66,55,JJ,JT.

If we do have air, the question remains whether bluffing the river makes any sense. Are there many hands in villains range with showdown value that he'd fold to a bet? I guess not, so I think we'd check the river with air, and we really do have a strong hand here.