re: Poker & $4 NLHE 6-max: BvB Flop TPTK on somewhat wet board
Firstly, I'd raise preflop; since I don't think it affects our ranges much, nor his calling range, and in absolute terms you're usually way ahead pre.
I'm guessing villain is quite aggressive post? If so, as played; I don't see a reason to bet the flop. You both have very wide ranges here, but I expect this villain to fire 90% of the time at this flop.
So, if you bet, he's folding most of his junk and possibly a few hands that you beat like 33,55,KQ, 4x. You'll get a call from hands you beat but don't fold like 66,77,56, 8T, TJ, diamonds. Anything that has you beat is not going anywhere.
But check-calling lets him bluff those hands that would fold to a bet and let's him "value bet" the other hands you're beating.
Now obviously hands that have you crushed won't fold to a bet, but there's no guarantee he raises with a set or 2 pair. He also raises with some of his semi-bluffs, meaning you only give up equity by bet-folding to a raise and a call doesn't always mean you're ahead.
So I think it's somewhat close but check-calling is more +EV than betting. Now your hand is going to look drawy + small PP plus 9J+. So this means that any diamond, 7, T, J,Q,K,A on the turn should scare him.
With the Ts, not much changes except some of his range most to the "beating you" column, and some to the "crushing you" column. If you check, he polarises his range by betting, which doesn't really narrow it down THAT much, right? So I think it's close between check-raising and check-calling. It's villain dependent. If you think he'll fold two pair, 444 or 888
or even 999, then go with the former, otherwise just call. You're still ahead of a considerable part of his range and you'll most likely have pot odds
If he checks (Ts hits your range a lot too), then you've a) pot controlled with a pair of 9s b) kept worse hands in c) minimised your losses when you're crushed and he's slowplaying d) and set up the river for a good bluff. All pretty good things.
If he bets, on the river, depending on the card, you'll have to decide whether to thin value bet, or check and decide.