Villain is opening about half his buttons. From the big blind he calls 46%, folds 46% and 3bets 8%, small sample 39 hands. Post flop he has been snap folding when he checks to me and I cbet, and has been donking 40% of the time with sizing 2/3 pot and has always continued on the turn 2/3 pot. I haven't got to show down vs this donk sequence yet so I'm not quite sure what to make of it.

Upon further thought I might fold this vs his open since we are not doing well equity wise against his range (we only have 42% vs a linear open range) and we will be OOP the rest of the hand.

I tend to check raise dry flops with air sometimes.

When he calls and I turn the top pair I'm not quite sure what the best move is. When I check raised the flop I was either planning for him to fold or for me to give up on turns. When I actually turned a "good hand" I wasn't really prepared and with time pressure I'm not quite sure I made the right play. I end up checking to him and then he leads for pot. Is this a fold, call, or shove? I'm sort of lost since his hand looks really strong that he called on the flop (on this dry of a board) and now wants to put a lot of money in. Seems like he has a strong hand that is trying to protect against the draws that have come in, or should I interpret this as a move villain would do with air?

Thanks!