Is there any way to save the shows as an mp4 file so you can watch it without the app, if that makes sense? I'mguessing the background would be blank. Or is there any way to save it as an mp4 file with a background of choice?

Not the animation files, specifically those ending *.vghd.
These are encoded to each specific user.
That is why it is not possible to share them.
The animation files ending *.demo, maybe. I am not sure if these can be viewed outside of the app. They are free clips not attached to any account.

The bonus videos can certainly be downloaded to any desired location, and in different resolutions. Make sure to check the resolution is what you want before downloading.
After downloading, they can be played in any media player.

I just tried to play this file "f1541_41201.demo" with Windows Media Player. Not possible.

This would be the ***** of the terms. File format protects the duplicating of the file for those, not paid for it.