First, Id for sure have a raising range on the flop. In my games donking is very common and depending on who does it can be anything from bottom pair to the nuts. So I'd keep that in mind and my raising range would be depending on who's donking.

On the river I'd almost alwys bet. Again I would often make an explo decission depending on villains tendencies. Some villains will often bluff if we check here and others will never bluff but instead call down very very light. As played now shove. 

raise flop, i like the river check now call. Raising range on flop can be all sets/overpairs and combo draws.

I think raise flop especially when fishy guy leads. Because he is likely to call you with worse and gives protection to your hand from draws and also gives you info where you are in the hand. I like raising the flop in this spot make it 550 or 575. Good reg is calling mostly a drawing hand or hand like KQ, QJ hand on the turn because he is raising most value hands on the flop like Q 10, 10 10, 77 and huge drawing hands like combo draws Jd9d, KdJd, Kd9d, 8d9d. I think turn bet is fine for value and protection he is likely going to fold his drawing hands like 89, J9 off. Which gives him a narrow range which has KJ and Jd9d or Jd8d. I think check decide the river but I don't know if folding would make you exploitable. As played the reg has KJ most of the time. That's how I feel because most of his just flush drawing hands would fold turn and or big combo draws would raise flop. I think 77s wouldn't bet so quickly on the river.

'As played the reg has KJ most of the time.'

I agree. I think we could find a river fold. Obviously in game I doubt anybody would.

IDK if we should do anything different flop and turn. It seems to me that the turn lead looks very strong. So so unlikely that you make it as a bluff, and so unlikly that you make it with worse than AQ. It doesn't necessarily imply that you should take a different line, but something to take into account for the river: when the bottom of your range is AQ and reg quickly bet 2K, it's very coherent for him to have the nuts. He would pbly play all KJ combos this way, so it's kinda tough for him to have enough bluff combos ; and as Najamq said, the huge and quick bet in this situation weight vilain more towards the nuts than a thin value bet with AQ or 77 (that he might raise on the flop).

As for the flop: obv raising has a lot of strategic merits. One of them is to clear the field on this very drawy board when multiway (vs wide ranges!). But you have to know what to make of BB's big lead. It looks very strong in a 4way pot, and my standard would usually be to flat, but some players would just donk pot with QJ, totally misplaying their hand ; so you need to have an idea of how BB is considering the game before proceeding. You said he was fishy, so obviously you have reads on him, or at least a feel, and that shall lead you to the right course of action.

Turn, I think it's important to value bet / protect your hand, avoidng it to be checked through for sure. But it still sucks because 2 out or 3 of your opponents can easily have KJ, and will not call much else with you having the Ad. Still not sure what I'd do as played. The hand is way easier to play if you raise the flop obv.

prty sure he has kj
what else can he rly be bettling like this with