Update!
I tried calling the DMV, but there was no way to talk to a person... at least I couldn't find a way.
Then I remembered that they actually do support via twitter too, so I got some answers. Here's the full conversation:
Me: I moved to FL. Can a car I own in CA have a FL address on registration? Please help! Thanks!
CA DMV: Yes, you may have an out of state address on record for a vehicle registered in CA.
Me: Great, thank you! And will I receive the registration and sticker in the mail at my FL address so I can send it to my mom in CA?
CA DMV: Yes, the tag and registration card will be issued to the address on the record.
Me: Thank you so much for all your help!
So there you have it. You can have a Florida (or any state) mailing address for a car registered in CA