Full Tilt Poker $1/$2 No Limit Hold'em - 8 players
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UTG: $88.10
UTG+1: $61.50
MP1: $302.15
MP2: $40.00
CO: $45.45
Hero (BTN): $200.00
SB: $211.35
BB: $235.75

Pre Flop: ($3.00) Hero is BTN with A K
5 folds, Hero raises to $6, SB calls $5, BB calls $4

Flop: ($18.00) 9 2 K (3 players)
SB checks, BB checks, Hero bets $12, SB folds, BB calls $12

Turn: ($42.00) T (2 players)
BB checks, Hero bets $31, BB raises to $75, Hero calls $44

River: ($192.00) 8 (2 players)
BB bets $142.75 all in, Hero folds

Villain is 22/9/2.5 over 240 hands. My three questions are
1) Should we be betting the turn vs. this type of player?
I think his range for calling flop is any K, flush draws and possibly (but less likely) underpairs to the king, sets, and K9. Against that range the underpairs won't give us value w/ a turn bet but may give us value w/a river bet. Also, we open ourselves up to a c/r, and we are not scared of the 4th diamond (which gives us the nuts). I think if we get 3 streets vs. worse K's then it makes it worth betting, but vs. this type of player when the flop was multiway I doubt it, so I think betting was bad on the turn, but would've been good if we didn't have the nfd. Thoughts?
2) How is our bet sizing on the turn?
I think betting too small can open us up to him turning an underpair into a bluff, but my guess is underpairs are a small part of his range. Thus, we want to get the most value from his K's, so should I have bet smaller?
3) Do we really have implied odds to call turn c/r?
Firstly, how often do we think he has a flush when he c/r the turn? Secondly, do you think he pays us off enough when we hit our 4-flush? I am not sure.

It looks to me like I totally botched this hand, and I should've not bet turn, or bet it smaller, and as played potentially folded to the turn c/r. Thoughts?

Wow this is bad, Alexander!

What is your reasoning behind calling the turn c/r? You don't have enough implied odds if you're folding to a brick on the river. Do you call if you hit a Ks on the river? What about an As?

He's not going to give you 3 streets with a worse King.

We need to be getting about 5:1 implied odds. After he raises there is 148 in the pot and it is 44 to call (we need to be winning at least 220), making it 192 going to the river. Thus if he bets or will call an average of about 40 on the river if we hit then the call is breakeven, which is pretty likely, so now that I think of it even if we are folding to any non flush river our call on the turn is probably not awful (not saying it is actually profitable), however that's not to say I still didn't botch the rest of the hand.