How? mavity is a force and hence is acceleration whereas the speed of light is a velocity. How can the two be the same?
All forces are transmitted by massless particles which travel at the speed of light (at least according to our current best theories of physics). Electromagnetic forces are transmitted by photons (which we can detect with relative ease), whereas gravitational forces are transmitted by gravitons (which we cannot yet detect reliably).
It's certainly true that we do not have a fully unified theory for mavity, and this is where a large proportion of theoretical physics research is focussed. But, unless our theories require a massive revision, gravitons will not travel faster than the speed of light. People tend to think of the speed of light as being a "cosmological maximum speed", which is true in a way, but the speed limit applies to
all information transfer - not just objects that have a mass.
Look into
force carrier particles for more info.
Anyway, to answer the OP's questions:
1: The behaviour would depend on the material interface between the two boxes. If we take an idealised scenario with a perfectly isotropic surface (i.e. it "looks the same in every direction"; no alignment of microscopic surface irregularities etc), then the top box will not migrate, since the box below it is exerting a periodic force which has a net force of zero over the period of oscillation. Since the bottom box is moving so rapidly it's reasonable to assume that the force will be enough to overcome the static friction, so the top box would not "move with" the bottom box. There would, however, be a whole assload of friction created, which would heat up the surfaces of the two boxes very quickly.
In a less ideal scenario, tiny differences in the surfaces of the boxes would mean that the force exerted on the upper box, when the lower box is moving "left", would be a tiny fraction different that that exerted when the box is moving "right". The effect of this would be a non-zero net force on the top box, leading to a slight overall movement in one direction (in the same way that a vibrating phone may migrate across a desk). Eventually it will move off the top box and fall.
To answer question 2 you will need to get into string theory, or some other theory of quantum mavity. Since these are not yet complete, it's of little sense to get into specifics
It's interesting the way you chose to phrase the question though... There are many scientists who believe that the event horizon of a black hole represents a very real boundary to our Universe, and that inside of the event horizon the laws of physics as we define them cease to exist (
Cosmic censorship hypothesis). In this way, you could argue that the original object does "cease to exist", but then you're getting into philosophy