Pre-flop : I call every time. I would only raise with AK, AA, KK and QQ (not all the time with QQ).

Unfortunately, you get 2 callers (they are likely calling with medium/drawing hands, because they didn't raise).

On the flop :

SB : He could have a hand and going for a CR or nothing.
Raiser : He doesn't have a good hand 95% of the time. Why wouldn't you bet if you have a set, TPTK, or AA? There's 2 players left( plus the SB) and there's a FLUSH DRAW.
You : You checked. I would bet 2400 here, you will take it down 50% of the time, be called by a Flush draw 25% of the time and called by a K 25% of the time. If I get raised or CR, I would fold. But you checked.
Checked behind.

What should you think. There might be someone on the Flush Draw, SB could have done a CR but MP1 and the player behind have nothing.

Turn K:
SB:Check
MP1: Bets 2400 - I still thinks he has nothing, I would Play WAWB if HU, but with flush draw out there and 2 other players, I would try to take the pot down. POt = 6600 and you have 10000. You have to at least raise to 5k. Would you bet 5k and then fold if raised? No, so I would go all-in. That has the advantage of making fold any draw, so you will only get called by players who missplayed their hand.

All-called.

River : Runner-Runner flush.

You can only chek this, and like you said call a medium bet by the last player to act.

What I think, is that player don't slow-play when there's a flush draw and 3 other players in a hand. So I would go ahead and give it a try on the flop.