Right. The mirror will simply fold the projection light path in the middle. If you place the mirror right at 1/2 the projector's throw distance, it will give you the shortest overall depth.
I don't think you will really need a special rack or frame, unless you need to move the setup often. The projector should be either shelf or ceiling mounted in the back room facing backwards, and the mirror should be placed vertically at a half the throw distance. The only tricky thing is that you will need a mirror that has a reflecting metal coating on the front surface, not on the back side of the glass (the latter will give you a bad chromatic aberration, i.e., a prism-like effect).
