Dont call this pre against short stack imo
AP-Flop raise is ok you are folding a ton of Ax
"Once he calls flop i feel he has has a 9 with a strong kicker and overpairs, maybe a set.
We pick up a a pair so no need for us to barrell in position so i decide to check back"-Would you barrel if you didnt hit a 6?If you would then 6 is even better hand to barrel then most of others,you pick up extra equity and you dont have SDV because his range is A9+
River is fine imo