Yeah, I agree with Bender. Alot of the games that did this also have a backdrop the image projects onto, giving it a more realistic look (like Space Invaders, Omega Race, Asteroids Deluxe, etc).
Any normal games which used a mirror, did so strictly for overlay purposes. I've seen games with mirrors on a black backdrop, but that was more than likely because the overlay portion was missing, damaged, etc... There is a benefit to doing this, however. In an environment with a lot of ambient light where the source isn't directly behind the player (like a big window to the outside), the monitor is more immune to washout, as it's face is deeply shrouded.
I think some of the shooting games may have needed it due to the close proximity of the gun.
Pretty much. Technically speaking, you don't need a lot of distance, as long as you can make the targets small enough to
appear to be in the distance. This is easy to do nowadays, because common resolutions in use are much greater and small targets can still be well defined. Not so much with the lower res monitors of that era, so they really needed to push them back
optically to keep detail in the imagery. One of the neat things about those shooting games was the fact that the monitor face actually appeared to be further away than was possible, given the physically shorter depth of the cabinet. That's something which can only be done by putting a large(ish) monitor way down deep in the base of the cabinet and using a fold mirror. It's also important to note that these mirrors were probably what is known as "first surface" mirrors (or at least they should have been). These are mirrors that have the reflective surface on the same side as the light they are reflecting, which eliminates ghost images, as well as the ~4% per reflected surface loss in brightness.