How often is 略字 (handwriting simplification) used? Is (handwriting simplification) used only in informal writing (like a diary or post-it annotations) or is it used in more formal situations (like teacher/professor blackboard writing or an essay)?

Well, there are many types of . <

Short answer: it might be used on blackboard.

Generally speaking, I don't think primary school teachers use them, and also unlikely secondary school teachers. I remember my history teacher in high school using for . But it was rather a memorable exception. As for essays, probably it does not make much sense because everything is typed these days.

Almost _the_ everybody uses and recognizes would be (with vertical stroke at the top) for see the link above. Including this one, there are a number of abbreviations that surprise nobody if used by teachers, but not so many.

Personally I think people (including me) just use characters deformed in their own way, not the clearly defined variant of those characters.