I don't like limping pre flop to open. I might be on board with it if you are 95% sure or more that someone after you will raise but there is probably no way of being that sure. Then that would lead to a limp raise play which depending on how many times you have done it or not done it at all will make your hand face up. You would be better off open raising in these spots to protect against the times your opponents just call behind.
As far as the flop play goes, your bet is way too small. BB called your raise so more than likely they have a little something and depending on the opponent that little something is enough for them to float you. If they have nothing on the flop anyway then you will likely not get any more value from them later in the hand and you are best off trying to get value on the flop where they could have a tendency to float (not sure about the opponent in this hand, also not sure about the suits of the cards on the board, what your suits were of the KK, stacks, etc. which all could make a big difference in this hand).
As played on the turn the ace could our opponent but it is also part of our perceived range as well. I would barrel the turn here especially since you bet so small on the flop. If the villain raises here then it is very likely that they are not
bluffing since we did a limp raise move pre that most players think is only done with a few
hands and doubt they would
bluff into that. betting here and then folding is not the end of the world since again if villain has nothing they fold and unlikely we would get any value on the river anyway and if they have an ace we can fold easily to a raise which might save us chips vs letting the villain bet on the river after another check. We can also perceive a call from villain on a hypothetical turn bet the same way we would look at a raise. They would not be calling us there two streets with the hands we should have based on pre flop action if they did not have an ace.
As the hand is played on the river, I think this is probably closer than you would think. If I was playing I would not call since I am nitty and probably just couldn't pull the trigger there for the call but villain could easily have seen (depending on how good they are) that the ace scared you and you shut it down after that and they see a spot to just take it. I would think they are bluffing here quite a bit but its not enough for me personally to make the call.
