Here is my analysis...
You raised deep stacked in early/mid to less than 2.5x. An average range for this type of bet, with your chip stack, in this position would be JQ(s),KJ(s),KQ(s),AJ(s),AQ(s), AK(s) and 66+
The BB, is also deep stacked and has you covered. It is in the early/middle stages of the tournament, so he is happy to call your bet cheaply and see a flop. I would think his calling range here could be any connectors, suited one-gappers, suited aces, AQ (s) and under (he's almost certainly re-raising AK at this stage) and any pair 10s and under.
: (988) 6
A (2 players)
Actual Action: The villain checks and you check behind.
The first thing you need to think about here (as with any heads up situation in poker) is your read on the opponent. Is he likely to float air and represent any draws that hit? Has he checked back top pair to the raiser in the past or has he led out with it? How likely is he to back down to a continuation bet? Is his calling range tighter or looser than average?
Against an aggressive opponent who has check-raised or called/floated many flops in these situations in the past, I tend to take the conservative route here and check the flop. Checking here against this type of opponent has a few advantages. #1, it keeps the pot small on a heavy drawing board when you have a made hand with little chance of improvement. #2, checking the flop often times will make this type of player bluff at almost any turn card. In these cases, you can call the turn bluff and check/call most river cards.
Against a standard opponent who I have little or no read on, I will almost certainly make a standard continuation bet on this flop. A few things can happen for me when making a cont. bet here. #1 The villain folds and I win a 988 chip pot. #2 The villain calls and helps to define his hand further or #3 the villain raises and leaves me with a fairly easy decision to fold.
I am fine with your decision to check behind here and would do so myself if the circumstance made sense.
Actual Action: The villain bets out 400 into 988 (roughly 40% of the pot) and you put in a 3x raise to 1200.
In this type of deep stacked heads up situation, I like to just flat call here for a couple of reasons. First, his bet is not so small that it lets him see the river too cheaply if he is betting a draw. (He is 15% to hit his flush draw on the river). Second, if we raise here to 1200 and he 4 bets we will almost certainly have to give up and would have wasted an additional 800 chips. Lastly, by just calling here we allow ourselves to keep the pot smaller and either call a small river bet (often times a bluff) or check behind and likely win the pot.
*Note that he did not bet out the flop on this hand. When an aggressive player flops a flush draw, they will often times try and take you off the hand right away or get you to check the turn so they can check behind and see a free river card. This fact is something to keep in mind when analyzing the hand and makes it less likely that he is on a club draw. (You are also holding the 10 of clubs, which falls into his range).
Actual Action: You bet 1,800 (into 3,388 or 53%) and the villain raised to 5,760 (about 3x)
The 5 of spades completes the low straight draw and the backdoor flush draw. His smallish lead on the turn usually indicates a hand like 89 or a K or Q high flush draw. I think if he had flopped the ace and checked, he would be making a more sizable turn bet here. His action on the river is consistent with holding 89 or the backdoor flush draw. If you don't put what is essentially a bluff in on the river, he will likely check behind any hands that you beat...