Any mixer that cooperates and listens to government demands shouldn't be trusted. Today the reasons could be that the money is stolen or hacked. Tomorrow, it could be because the owner is black, white, Jewish, Muslim, in the opposite party, anti government, too liberal, fascist, pro gay, young, old, American, Russian, Iranian... No censorship at all. A little bit of censorship opens the door to keep raising that bar in the future little by little.
Wasabi's coordinator zkSNACKs works with an unnamed blockchain analysis company to surveil and identity UTXOs they consider "unwanted", "dangerous" or simply "tainted" in their own definition of those words. If the company tags certain UTXOs as such, they will be banned from coinjoining.
Quote
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.PS. I'm looking for mixing services to come to this thread and respond so that my confusion regarding the same gets cleared. I mean, if they're interested to reply here... 
In that case, I would make them aware of the existence of this thread. PM the forum representatives you would like to hear from. But remember, it's all about image. None of them will tell you they will cooperate with any government investigation. What would really happen is something only they know, and it's not for the public.