As an exercise in reviewing the music, once you identify the note/chord whether you write it down or not, as far as I'm concerned, is a personal thing.
If you are a visual learner, it probably will help, otherwise, I see little value in doing so. Other than writing it's name on the sheet music will help you remember the note name later as you play the music.
Yes, to notations in the margins as to how this section should be played, etc.
Of course, IMO.