I personally like to check the flop on middle pair. This goes against conventional poker wisdom, but I feel like I can usually get a sense for the kind of cards people have, if I check it out to them and see how they bet. This kind of play works best if you get a read on your opponents not having very strong hands, even if they bet, and then you retake control of the betting on the turn.
Why would you reraise with A10 on the button? You have position on the guy, just call. He raised UTG which usually is a sign he has a pretty good hand and not trying to steal the blinds in a cash game. Raising with A10 off was a bad move to begin with, and based on this flop and his raise I play middle pair very cautious here.
I'll check-raise if its checked to the player in position with 4 or less people in the pot If I think they are just trying to make a stab at the pot, sometimes double and sometimes a 3-bet, this goes especially for LAG's that have position. If they are tight, the chances I do this decrease but I still do it, I'd probably be more inclined to check call a tight person and check call if I'm getting pot odds with outs to a better hand with more than 4 people in the pot. You could check-raise the position bet, or check raise an out of position bet by an LAG'r thats known to take stabs at the pot.
Know one thing tho, if you check-raise the flop you need to follow up and bet the turn.
It's always nice to see cheap flops with middle pair. I hardly ever limp in early position since it could be a waste of money if raised big behind you....but then again sometimes an early position limp looks like a big hand so it can be played that way against good players who might suspect AA or KK limped early...against donks its never a good early limp cause you will never be able to bluff them or push them off top pr to the board. GL.