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Cleanup: split up `ed_screen_context()` into separate functions

Refactor `ed_screen_context()` to call separate functions, instead of
having the entire functionality in one function. Each function now only
retrieves the data it needs from the context. Furthermore, some string
comparisons are removed.

No functional changes.

Reviwed by: brecht

Differential Revision: https://developer.blender.org/D9090