1. Do mixers know where they're getting a deposit from?
The mixing teams are coders and developers they know how everything works so it's likely they know how to trace addresses this is mixing business and their business is privacy anonymization of addresses
2. If yes, will they cooperate here with the officials and let them know about the deposit?
There is not one answer and the answer is not general nor they will give the same answer but they know that this going to come while they are in operation, and some will likely cooperate because they are not in the business of conniving with hackers and scammers,
the easiest way to close your business by authorities is to help scammers getaway
3. If they do, how will they stop the scammer from moving his funds out of their mixer?
They have a way to do that they are the developer of the platform, and developers always have something kept secret from their users this is for foreseeable circumstances.