Interesting results. The article states that Faraday cages are no directional. My physics 101 understanding of them was otherwise. You show they work by integrating over the surface of the cage and saying that's equal to the charge within the surface. But you don't say it's equal to the charge outside. Indeed it couldn't possible, since that charge could be light-years away and steady state would never be achieved. N…
If you consider the compactification of R^3 by a single point “at infinity”, an electric charge inside the surface is the same as the opposite charge at infinity (this is just Gauss’ result). In that sense, “infinity” acts as the opposite charge, does it not? I do not understand your question, otherwise. (Not a criticism, just an interested comment which may be very wrong though).
This is a mental operation, not physically possible. The difference is real: for example, charge inside a perfect conductor cavity generates electrostatic field outside. A charge outside perfect conductor cavity does not generate any static field inside.