I plan on using flat mirror projection, among other methods, to view Tuesday's partial eclipse. How do I calculate the distance my flat mirror has to be from my screen in order for it to cast a sharp, bright Sun image rather than its own shape or a blurred image?
One time I put a "pinhole" in some paper, it was roughly 5 or 10 mm and roughly round (I think I just pushed a pen through it) placed the paper over a flat household mirror, inclined the mirror so that it reflected the Sun through an open window and into a darkened area indoors where the resulting image was projected on to a white wall.
To calculate the expected performance I'll suggest using the technique of similar triangles (see also Similarity system of triangles for fun) and the small angle approximation and take the limit of the distance of the Sun being infinite.
If the angular width of the sun $\theta$ is say 0.5 degrees, that's roughly 0.01 radians. That means that the width of the sun's disk at a distance $L$ from a pinhole or any small aperture (transmission or reflection) will be the product $L \theta$. If $L$ is 10 meters, then the height of the Sun's disk will be about 10 cm.
To get enough light through your aperture, it needs to be big enough. I recommend you have a series of apertures of different sizes available or some manipulable material like a simple sheet of paper where you can poke various size holes in it until you get satisfactory brightness.
Since rays from the Sun are essentially parallel, "fuzziness" of the image will simply be the aperture size $w$, independent of distance.
So for a 0.5 cm aperture at 10 meters, you have a 10 cm solar disk which is "fuzzy" by 0.5 cm.
Solar eclipse viewing via apodized mirror projection: