a) So he bets $1.25 into a $1.75 pot on the turn, you mean? Most likely call. If he's bluffing a second barrel, a small bet on the river when he checks to you (three barrel bluffs are rare) will be a better way to bluff than raising the turn, unless he's really looking for an excuse to fold QQ on a board like this, which I don't think he is. He's obviously on the lookout for the flushdraw, but you have other outs to the nuts as well, so you'll most probably be able to extract the required value out of your hand when you hit, on average.
b) I bet 100% of the time. If it looks like he's giving up, he probably is. Bet to fold out 55 and AJ and whatnot. Lots of hands beat you but are very uninterested in continuing; take advantage of that.
What to do if he check/raises the turn? Pot odds
would dictate that for me. If he check/shoves, I fold. But I think he check/raises the turn so rarely that it's still much, much better to bet when checked to and risk forfeiting equity than it is to check back.