You should be fine to just move over the files and export the db. You will need to update the site specific settings in the db inside of the wa_settings table. In here for both PowerStore and PowerGallery the local root, remote root and root URL are listed. This is probably why your gallery did not work for when moved into the root of the site.
You will need to check on the wa_settings table for these values and ensure that they get updated to the correct values for the target server. If you have a look at these values you should get an idea for how they are currently setup and how you should update them.