I seem to be making myself very exploitable in a lot of spots... paired boards especially. Here's an example of a hand where villain is probably betting R 100%. Is there nothing else I can do here except check some top pairs, etc.?

Also, any advice on bet sizing and playing these boards OOP would be super appreciated... I'm sure I'm leaving a lot on the table here.

6max 50nl

MP: 100bb
HJ (Hero 66): 100bb
CO: 100bb
SB: 100bb
BB (villain): 100bb

Preflop (1.5bb):
Hero raises to 3bb, BB calls.

Flop (6.5bb) T33
BBchecks, hero bets 2bb, BB calls.

Turn (10.5bb) 7
BB checks, hero checks.

River (10.5bb) J
BB bets 5.25bb

I would not be checking TP to offten OTT, since you are IP, you are going to improve on some rivers with your checks, + sometimes bluffcatch, so he can't exploit you

Is that because we don't have (as) many value hands here? What can we take into consideration when bluff catching otr?

If you are worried about being exploited then just run your range through MDF and see if this should be a call or not. As long as you are calling down with the right frequency you will be fine.

These kind of boards are weird. I think you can c-bet them fairly often (with any pair or backdoor) if you go with half pot, but if you bet small you should counter-intuitively bet at a lower frequency and with a stronger range, because villain will be check-raising quite a lot. At equilibrium, I'm not completely sure if 66 belongs in a c-bet range on that board, but it might be fine if villain is hardly ever check-raising. You're not really getting much "protection" if villain will float with two overcards vs that sizing.
As played, I think you have to check back the turn and then fold the river. You can't defend with everything you c-bet with, and 66 is particularly weak in this spot. You have better bluffcatchers than that. (Call with 7x, for instance).

Call.

How much can we really checkraise in villains shoes on this texture?

I haven't got this exact situation in my spreadsheet, but for BBvBTN, Snowie has a check-raising frequency of over 30% on QQ9tt, J22tt, TT7tt, 995tt. I think it goes down a bit on rainbows like this T33, but there's still a lot of stuff that prefers to x-raise instead of call or fold. It's one of those boards where - at equilibrium according to the solvers - what looks like a leveling war breaks out, because nobody has anything, so you can have ridiculous situations where the BB 4-bets the flop with 54s for lolz, while shouting "I flopped trips. You gotta believe me!"

fwiw Snowie says check all tens and underpairs and bet our FH, quads and overpairs... Of course, for a 3x open Snowie also says we never have 6s in our range here. *rolls eyes*