It is starting to get fairly cold (20s-30s) where I live. When I start my 2005 R50 (Automatic CVT), it starts just fine. I back out in reverse and when I put it into drive, the rear end shakes and vibrates and the RPMs get very low. Today it even killed itself. It's really weird. Any thoughts? (It only occurs for maybe 1-2 seconds and the RPMs drop to 600)

 

Mine was shaking and I needed new spark plugs and a new spark plug coil - problem solved.

 

It's a violent shake that only lasts for a second or two. I changed the plugs about a month ago

 

I fear it is your CVT tranny...
You might want to read up a bit...
But the basics is the CVT was very troubleprone, and resulted in a class action lawsuit BMW settled....
Some folks HAVE done OK with the CVT....gentle driving and REGULAR fluid changes are a must...