SOLUTION:
If you are having problems restoring different backup images to iPads, look no further. After you create a backup image, and before you try to restore it to more iPads, QUIT APPLE CONFIGURATOR. Completely exit out of Apple Configurator, and open it back up, and restore away! Between different backup images, whether you are creating or restoring, completely quit out of Apple Configurator.
For some reason, Apple Configurator has an issue with loading the default Supervision profile back onto the iPadsbetween different backups. Whenever I had the problem where my account passwords wouldn't transfer over, or my restrictions passcode wouldn't stick, I could see in the General tab that it no longer said that it was supervised, even though it was. If I then refreshed with no changes made, the Supervision profile would then magically pop up in the General settings. Quitting Apple Configurator was the only way to stop this from happening. It would be awesome if this was fixed...