I'm new to the forums, but not new to poker. Needless to say, take my advice and thoughts as is, this is just how I would play it.
1st Street - FLOP
The board isn't particularly draw heavy, and you have his range crushed on the flop with the exception of hands such as 22, 77, TT. This is just a small portion of his range. Overpairs, and pocket pairs is what I have him on, so I'd bet 2/3 pot here, around 1.80 - 2.00 somewhere in that ballpark. I think your bet sizing was too low there, I think you were trying to make it cheap enough for the whale to come along in the BB?
2nd Street - TURN
The queen of diamonds doesn't hurt your equity here, no draw type hands made it, I actually think its +EV for you because it opens up some flush and straight draws that the villians will call you on. Unlikely you will see QT here show up, but it is possible. I think you have more then enough equity to bet here, bet size of 5.00.
I don't think your check is a very good +EV move, you have a hand that you need to be value betting and the board isn't THAT scary. Unless you planned on check raising the turn? Which is fine against an aggresive player, but you said this player was passive, so to me that's a -EV move.
I believe he is overbetting this turn for two reasons:
1) He has air or a marginal made hand with a draw opportunity, and seeing your check on the turn he decides to over bet the pot to win it. Which if you planned on check raising the turn anyways isn't it an instant ship it? I'm more scared of a value'd size bet on the turn enticing you to call then some enormous overbet.
2) He has a set that's he been slow playing and seeing you slow down he's trying to get his money in.
As played, I'd go with option 1 and ship it. I think you will see him show up with pocket 88's, and pocket 99's here a lot, or some kind of JTs type hand that connected on the board and had some draw possibilities.