Is there any way to automatically sort all apps across all screens?
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.