I'm good with the flop check/raise if his c-bet stats indicate he can fold. Our middle pair has pretty much zero showdown value out of position (there's no way we get to showdown profitably). Just keep in mind that our nine is pretty damn dirty. But this isn't a great hand to play for implied odds
(just call, try to make our hand cheap) because we're out of position, and our flush/9 outs are scary & obvious.
Betting the turn is okay I suppose, given the T makes a lot of draws, and it's fairly scary.
Shoving over his turn raise is pure spew. He's clearly never folding. Just call, make your draw or don't. If you don't, you save a few bucks.