As soon as I paid for the car they sent the title with the lien release overnight.
Yes, you keep the plate in Arizona, but it's illegal to put it on the car before you've actually registered the car.
If you go to
http://www.servicearizona.com/ you'll see that you can pay $1.00 and get a restricted three day permit to let you move the vehicle around for registration, emissions testing, etc.
There's also a way to get a refund for remaining credit on the BMW plate if it's over a certain amount. That's also discussed on servicearizona.com.