I still have trouble completly understanding your rational in the T9 hand,
PF:Hero delt T9
Villain raises to $3
Hero calls $2
Villan bets $3
Hero raises $6 to $9
Villan Calls $6
I think this play has little EV because when he calls or raises he most certianly has better. If hes raising 40% of his hands, Poker Stove says that range is: 44+,A2s+,K2s+,Q4s+,J7s+,T7s+,97s+,87s,A3o+,K7o+,Q8o+,J8o+,T9o. I think that range is more like 22+,A2+,K2+,Q6+,J7+ with some connectors in there too.
The way hes playing i think he calls any pair of kings(k3) or nines(j9), JJ, QQ maybe 33-88 and raises 22, 99, KK, AA, K9 and folds everything else (86, Q5). He may slowplay a set here but that would make this play even worse.
Based on his previous play I dont see any reason hes going crazy post flop. Yes when you had AQ he 3bet shoved on you but that was pf. I may be wrong but just because someone showed a crazy move (once) preflop doesnt mean their going to go crazy postflop. So i am rather certian he will not 3bet you here with AQ or 55. I think the last hand where he has 45 on the 234 board shows that he needs at tp+ draw to raise. Also he check raised, not bet/3bet. Also when he check calls how do you know he doesnt have JJ or QQ?
In the long term wont you win more just check calling the flop, letting the turn check though and making a value bet on the river.
The main thing Im trying to figure out here is why it is +EV to check raise 2nd pair here and how and under what circumstances would i want to add this move to my play.