Is there a way to add an animated object in the background of the home screen on Android? For example, an icon that would move in the background, without creating a livewallpaper to contain this special icon?
Yes, you can create a home screen widget, however it is not recommended. From CommonsWare:
Do not animate app widgets, unless you write you own home screen app.
More on this topic can be found on: Is there a way to animate on a Home Widget?