Question 1: Should I cbet a lot of this flop? I'm OOP but is pretty dry.

As a general rule when ranges are still wide enough you would want to check a decent amount on flops OOP. Unless those flops connect really well with your range vs villains. Or your range advantage is still a significant portion of your range vs his.

Question 2: In those spots, with no reads on villain, should I just double barrel often or not?

Even when we cbet entire range on flops we almost always want to select which hands we cbet on later streets. If you consider a hand such as AK it becomes tougher. Are we getting better to fold? Likely not on this street but might set up for the river. Are we getting worse to call? Likely not.
So if we would bet AK it's as a bluff. Then you have to wonder if AK is low enough in your range to be turned into a bluff? I doubt it. It's more likely that AK is still a strong hand that might even go in our check/call range.

Seems well played.

If we dive deeper, there not many range bets in deep poker. Board is pretty dry and would be better for you as stacks go shallower (less spr). Opponent holds advantage in sets here and pretty significant since 44/22 not in your range. Most of his range composed from decent hands on this board which like to face small size and have an easy continue.
I run quick sim and it shows that you have ~54,X% equity vs his range and around 108% EQRealisisation. Which can tell this board is pretty good for you (all for 100bb). Half pot and bigger bets are proffered, vs small bet villian folding range ~7-10% which is normal number.

P.s. villain turn and river checks not good, missed value