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.
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.Of course BitMixer wouldn't do this in the past, as it gives me enough to destroy their reputation. But now that they've closed their business, what stops them from giving out signatures at request to be used at lawsuits from big money launderers?
If I go full paranoid, this could even have been a reason to close the site: save someone who's being charged with something. My point is: this signature isn't worth anything for a judge.