Apologies in advance for complete ignorance of optics (beyond conic sections)
How is the size, distance, and angle of the mirror used here calculated? The mirror is I believe the open flap in front of the projector.
That sort of thing is done by simple ray tracing. It is pretty easy to compute the path of the rays at the edge of the beam by tracing from the edge of the object. In a projector like that, the object is the LCD screen inside the projector. In practice one generally doesn't have to do the ray tracing by hand though -- information about the required size of the optics is readily available in the same software that is used to design the projector lens.