Is that true? Film projectors cut off the edge of the frame? Is that a regular occurrence?
The camera aperture and projector aperture are different, yes.
For example see here:
https://en.wikipedia.org/wiki/35_mm_fil ... ifications
So directors who film in film film knowing in advance that the edge of the frame will be cropped?
Yes, see this for exposed area vs the framing for spherical 1.85:1 for example:
And what do you mean by "the edge of the frame, with hair and other crap"?
The edge of the frame is not particularly sharp and can be uneven.
Examine the edges this Kodachrome slide for example: