To answer your first two questions, yes, it is a known construction (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 approach thermal equilibrium. If you're asking about comparing an output integral to an input integral without such physical considerations, then you get an imbalance. I'm a little rusty on this, but I've been told that thermodynamics as a theory cannot be derived from physical principles without some quantum mechanical input. In particular, conventional notions of entropy seem to require some kind of finiteness on microstates.