i like ur thoughts preflop. flop is indeed ok to cbet. like ur size. turn is really interesting. the question here is if he wud still call kx or even 9x and hands like 66,77,88,tt,jj. i mean u have a clear valuebet here and thats obviously played correctly. but u should ask urself if sometimes c/c is a good option in that spot. i dont think his range is strong enough at that point to call the turn and the river. because looking to ur turn bet it commits him somehow.
i sometimes play c/c on the turn, because ur range in his eyes but be insanely strong if u keep betting on the turn. but if u checks here he might sense weakness and bets himself with a hand u wouldnt have gotten any more value from. the board is a little bit scary. if the 9 was a 5 i wud recommend u to check the turn + there are few gutshot + fd combos possible for him.
the is tough. if he somehow has called ax on the flop ure only splitting right now. hard to believe he has many 9x's in his range besides k9s and a9. most of the 9x he shud fold on the turn + its not possible for 9x to have a fd on the turn. so i think he never ever calls 9t here for example. if u think hes calling a king here for sure bet. otherwise c/shove is pretty nice here. hes probably not aware of calling draw for odds etc so he can still get to the river with a fd for example. that fd wudnt pay u any more chips tho. so c/shove wud be best line in this spot. depends on the game flow so far.
wud be interesting to see that range he has on the turn. i mean ure obviously playing bet/call on the turn so ure value betting. depending on his turn range u can calculate if its better for u to shove the river or to play check/call.
i think villain had a9 anyway

. thats why im asking for a range in the end so u can think about if valuebetting is good here. that also might open the alternative to c/c the turn to keep bluffs and semibluffs in his range ....