When I uninstall ADE should I also remove "My Digital Editions" file or at least remove everything in it?
You can, but it doesn't matter. If you do, make sure you have a backup of any bought books it contained.
After a new install, you may need to deregister (ctrl-shift-D, cmd-shift-D on Mac) and reregister.
The uninstall sometimes leaves incomplete vestigates of your old registration around.