For a second, just think that the scammer keeps the coins in his wallet and doesn't move them for some time, however FBI and other major authorities are aware of this hack and they give orders to mixers that if a specific address is provided to the mixers (if they know exactly on which address the hacker has kept the coins), and they ask them to immediately report them about the same,
So how do Authorities go about it?
Like "Hey Yomix, please do not mix coins coming from this address, if you refuse to comply, we know where you are and we shall arrest you"
I think the fact that almost all the mixers around are not registered financial entities or under any regulatory board, they are not answerable to any authority except maybe their customers.
The only choice authorities are left with is to investigate and crack them down.
1. Do mixers know where they're getting a deposit from?
They don't need to, except possibly with the controversial wasabi wallet. I understand they can block certain inputs from "blacklisted sources.
2. If yes, will they co-operate here with the officials and let them know about the deposit?
This is what exchanges do, highly unlikely for mixers to do such a thing if you read their privacy and no log polices. How do you promise people privacy but then reveal private customer information to the Government. Then that's not a mixer.