Is the plugin piece in the dashboard just the visual interface to edit the tables in the db? Should I expect to be able to deactivate the plugin but leave all the other code and tables in place and have it still work?
No, the plugin needs to be network active or installed in the mu-plugins folder.
Are you importing content from the same domain you are mapping? If so you might be messing up the site by trying to import content from itself. You should do the imports before you map the domain.