I would generally 3-bet A9s pre.

With NFD, you can check/raise or check/call the flop. Both are fine. Since he bet fairly big on the flop, I would prefer check/call. Because if you get 3-bet on the flop, you're in a tough spot. You probably have to call, and then if you get a blank turn and he jams, you're not getting the right price. Getting it all-in on the flop is also not good, because you probably only have something like 40% equity in that scenario.

If you check/call, leading out on the turn can make sense. Something like 2/3 to 3/4 pot maybe? Checking is also fine.

Pretty much every action is mixed for this hand if you put it onto a solver so it's difficult to say what's the best best way to play it.

I think you played it fine. I'll sometimes 3b PF, and on the flop I'll sometimes call sometimes raise.

According my stats on this level the field is generally slightly over folding against aggression so I take mostly the aggressive line with draws to make as much use from it as possible and take pots away.