I think if you bet bigger and get shoved on your toast so I'd prefer a smaller bet and call a shove more than a big bet.  However above those options I'd prefer to bet fold smaller.

Even vs super wide gii range you barely have 30%

On the flop I'm betting bigger than $5.50 before I have a reason too do anything else. (And I'm bet/folding very comfortably with only 6 (unclean) outs to T2P.

Not sure about turnplay, checking back seems like a line which can't be bad. Bet/folding also strikes me as an OK play. When checking back I'm probably calling a bet on
2, 3 and 8 rivers
Any river which gives us trips or two pair (but doesn't complete the flushdraw)

vs unknown, isn't that better to flat preflop rather than 3bet? we are not gonna get paid off on A high board usually.. I guess this hand could be the top of my calling range. turn play is interesting, either check/reevaluate otr or bet 12ish and fold to a raise.

I feel like flatting pre is better than a 3bet here.  You should have plenty of better AKJ* combo's to 3bet, an you need to have a few in your flatting range.  This one seems weak enough to where it's better off used as a call.

Postflop seems fine as played.  Bet/folding this combo seems fine.

What worse hands is villain gonna call is on the turn if we bet? 

I think both preflop plays are ok. We protect against a FD or AQxx stuff on the turn. I dont think he will shove AXxx+fd on turn now so his raising range would be wheelstr8 and some set or aces up. Thats why betfolding turn is the best play here imo.