Can I prevent home screen shortcuts from getting deleted or moved around?
There's no built-in functionality for this, but there are a handful of third-party launchers that support locking the desktop. Off the top of my head, I know that both Apex Launcher and ADW.Launcher do.

For Apex, simply double tap on an empty space on the home screen and then select "Lock Desktop" to prevent icons from being added or removed.



On ADW.Launcher you should be find a "Lock Desktop" option in the settings, which you can access from the overflow menu in the top right of the app drawer (Apex also has a settings menu there).

Other launchers that have the “Lock Desktop” option:


Holo Launcher and Holo Launcher HD
Lightning Launcher