I want to make sure that I make a complete backup of my MC database on other media (CD) so it can not be damaged.
File > Library > Back Up Library... creates a zip file that contains all database info. It can be restored (File > Library > Restore Library...). When a library is restored it replaces all data in the currently active library. You can also create a new empty libary with the Library Manager and restore the backup file in that library.
Where are all the key files located (I used all the defaults when installed)
On Windows XP:
C:\Documents and Settings\[your username]\Application Data\J River\Media Center 12... (you may need to set hidden folders visible in the Windows Explorer options.)
On Vista the location is slightly different. Someone posted recently the exact location.
The database files have a .jmd extension, but you don't need to backup them separately if you use MC's backup tool. The default cover art location is a subfolder in the same place. The cover art files are not included in a library backup zip file, so you may want to backup the cover art images separately. MC creates also automatic library backup files. You can find them in another subfolder in the same location.
and it would be helpful if I could turn off the auto update feature immediately when I boot MC.
Tools > Options > Auto-Import > Auto-Import Options > Auto-Import Folders > Options > Untick the four tickboxes. You can disable these options before closing MC and removing the drive.
Can I still work on the database files while I have the drive shipped off to be fixed?
The library files that are on the removed drive will remain unchanged if you disabled the background import and maintenance options. They will be offline until the physical files are available again.
Be aware that usually broken drives are not fixed. If the drive is under warranty you will probably get a new empty drive.
Can you advise?
I hope so.
Edit
Also what MrHaugen said. He posted while I was composing my answer.