Does iOS 6 support Live wallpapers? Are there any apps out there?
Native iPhone iOS (un-jail broken) does not support Live wallpapers (as in Android). There are numerous apps available for jail broken iPhones in the Cydia app store. An example would be vWallpaper. Obviously you would have to jail break your iPhone to utilize a Live Wallpaper app.
Unfortunately not. It might be possible if you jailbreak your device, but it's not a feature of stock iOS, and App Store rules prevent third party apps from changing the wallpaper or running in the background, which would be necessary for changing it on a schedule.