To answer your first two questions, [yes, it is a known construction][1] (physics education journal, 1973).

If the mirrors are made of ordinary matter and the photons have any sort of wavelike properties, then the whole setup (including the mirrors) will achieve thermal equilibrium.  If you're asking about comparing an output integral to an input integral, then you get an imbalance.


  [1]: http://scholar.google.com/scholar?q=%22perpetuum%20mobile%22%20%22ellipsoidal%20mirror%22