1. Copying the war is your easiest option. 2. 2.0.1 was an Enterprise bug fix release, so you can get it if you are an Enterprise customer. You could also check out the latest code to get the fixes and any new functionality from 2.0 (see our wiki for how to do that)
We use a separate MySQL database with Alfresco. Does upgrading in the way you suggest (copying the new WAR file) leave the existing data intact and does it upgrade the data model (new tables) if required?
Assuming you did not modify the previous WAR file (unlikely) then yes that is correct, copy in the new WAR file. When the server starts it will use the same DB and will automatically upgrade the schema and apply patches as required.