It's the authorities responsibility to provide proof the funds were obtained nefariously.
Not in all countries.
Here it's your own responsibility to proof you've obtained for instance cash legally. Crypto is probably treated in the same way.
Wait, what? That's contradictory. How could it be signed by your service with an invalid signature? This sounds like a convenient excuse to deflect the blame.
I think it just means that there is a signature, possibly lifted from a real LoG but it's not a valid signature for the fake LoG.
That makes sense, I'm pretty sure not all users actually verify the LoG (especially since some users don't even bother downloading it).