BB vs Button

[Gene2x]

I was reviewing a recent session in PT when I came accross this hand- a confrontation by the BB and Button (I'm neither of the two). The BB has AdQc and the Button has 9d9h.

Party Poker 0.50/1 Hold'em (6 max, 5 handed) FTR converter on zerodivide.cx

Preflop:
1 fold, MP calls, Button raises, 1 fold, BB calls, MP calls.

Question: Shouldn't the BB have raised to try to knock out the MP player?

Flop: (6.50 SB) 4, Q, K (3 players)
BB checks, MP checks, Button bets, BB calls, MP folds.

Question: Would it have made sense for the BB to bet first to see where he's at instead of checking to the raiser?

Turn: (4.25 BB) 4 (2 players)
BB checks, Button bets, BB calls.

Question: The pot is offering about 5:1. If he's behind, he needs another Q or an A but has to discount some of them because the Button could have KQ or AK or might be betting a draw. In that case the BB has maybe 3 outs and the pot would need to be offering 15:1. Is this such an automatic call by the BB? Forget about reads on the Button. I think his bet is mandatory at this point regardless of what he has or type of player he is.

River: (6.25 BB) K (2 players)
BB checks, Button checks.

Question: Is checking the river correct (for both players)?

Final Pot: 6.25 BB