Note: I've never used BitMixer so I won't miss it's services. One of the reasons for not using it is that the coins I get back are most likely from someone who's trying to hide worse things than me

When you used Bitmixer, they would give you a "Letter of Guarantee" (mentioned
here) so that you could prove that you sent funds to their site (signed message from their address + signed message from you = proof).
If the authorities came knocking, you could just prove to them that you sent your funds to Bitmixer, and therefore that the tied coins were not yours.
I've seen it, but I don't think it proves anything as it still requires the authorities to believe BitMixer is honest.
For the deposit, it certainly doesn't require that. The point is for them to sign a message proving that the address you're sending to is owned by Bitmixer. So as for the funds coming back to you, you can sign a message proving that you own the address that received a transaction of the same amount you send to Bitmixer, if that makes sense.
After that, the authorities can either believe a mixture of cryptographic proof and completely obvious reasoning, believe that Bitmixer stole your money and you just mysteriously received a transaction from a criminal at that time, or believe that you own Bitmixer. I'd say the former one is the only logical way that they could go.
I doubt a judge would believe evidence created by the anonymous website that you claim laundered your money into coins wanted for something you claim you didn't do.
I don't see why not if you provide evidence that Bitcoin's signed message system is sound and you verify Bitmixer's signature.