2. If yes, will they co-operate here with the officials and let them know about the deposit?
3. If they do, how will they stop the scammer from moving his funds out of their mixer?
Since the mixers are operating in the gray area - let's be honest, at least in theory they protect privacy of both honest users and possibly money launderers too - they may simply not care where the money comes from. Of course, this is probably the official reason some mixers are targeted by law enforcement too (and it's better to honestly say "I didn't care, so I didn't know").
On the other hand, a mixer is just a private business as any. The mixer owner may take certain steps without announcing that (from locking and returning the stolen funds to keeping track of the transactions), but, since such things can hurt the image of a mixer, we will never know a mixer's real stance on that.
So I think that you're expecting too much.