Is it possible to use a GIF as my wallpaper?
There is no native way to do this, as Android simply does not support it. Any wallpaper which is not a live wallpaper will be displayed as a static image. Specifically, Romain Guy notes in a comment on the Android issue tracker that all regular wallpapers are saved as PNG images after cropping. There is no way to change this behaviour aside from modifying the Android source code.

There are some third-party apps that will allow you to use an animated GIF as a wallpaper, such as AnimGIF Live Wallpaper. The downside, though, is that I believe all of these will be live wallpaper applications, which sort of defeats the purpose.