Is there a tool or an app to automatically sort home screen icons?
Is there any way to automatically sort all apps across all screens?
There is a way! If you go to your Settings > General > Reset, there is an option to reset the home screen layout. Pick that option and all your apps will be rearranged in alphabetical order. But you'll lose all your folders and maps...
No. Apps are extremely limited when it comes to modifying anything outside their sandbox. You might find an app capable of sorting the icons if your device is jail broken. But there definitely isn't one available on the app store.