Seems wp, checking flop/turn is pretty bad imo since he can call with plenty of worse hands on both flop/turn+most of his hands that do stab will call your bet anyways. I also expect him to check back a good portion of his overcard hands so it's nice not to let him realize equity for free with that portion of his range.

Turn is very tough, because villain can have all the sets, and we can't have any of the sets. I would probably just check here, because if we bet, villain has a great spot to raise us and we're fucked. Villain has tons of nuts in his range, I would play my range passively and carefully.

Then again, it's NL50, he probably doesn't have a clue. Bet is fine, versus a better player I'd check for sure.

I'd potentially check flop or bet smaller given that villain can have all sets here and we only have 1010 along with the fact that our squeezing range is probably fairly high card heavy on this board and he has a bunch of the mid pairs. Checking the flop also protects our AK type combos from getting run over on this board. Once we do bet the flop and get to the turn I'd definitely check. On this turn villain picks up more combos of sets, his bluff catchers pick up equity and I honestly doubt this player has much 10x in his range once he calls the 3 bet.

Very interesting situation and comments also.
What do you think about a preflop cold call?
I mean, as James say, this flop/turn is not the best for getting value. But! a Q/K/A high board texture is also not good I think.
Actually offsuit broadways, sc-s, Axs are better for squeezing, aren't they?

I would x/c either the flop or the turn.