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.
Tongbu can do this.
It has a feature called "Smart Icons" in its icon manager.
It's actually very extensive the options it gives you. You can even chose to ignore existing folders as well as the first, second and third homescreens.
Even select what categories you want it to use.
I just use it in Sandboxie along with the Run as admin option (which is in sandboxies window). Everything works fine with no ill effects.
This works in all versions of iOS except for 9 as far as I know.
Part of the app is in Chinese and i think there is only a windows version but im not sure.