Renaming
A custom label can be renamed, and it stays attached to everything carrying it. Nothing needs reassigning.
Screen labels cannot be renamed directly. Rename the screen and its label follows.
Deleting
From the label dialog, which has a delete control against each label.
Deleting removes it from every screen and playlist carrying it. Any playlist that was reaching screens only through that label stops reaching them, immediately and quietly: no error, no warning on the playlist, just screens that stop showing something.
Check before deleting
Search the label name in the screens list to see which screens carry it, then in the playlists list to see which playlists target it.
If the second list is not empty, deleting the label takes those playlists off those screens. Retarget them first.
Tidying up
Labels accumulate. Campaign labels from last year, a site that closed, a typo nobody removed.
Two rules make the clean-up safe. Delete a label only when the playlists list for it is empty. And treat a label carried by no screens as a candidate rather than a certainty, because a playlist may still be targeting it and waiting for a screen that has not arrived yet.