Also, if event horizon scales with mass, does matter inside the event horizon have a specifiable density?

Thanks for a very intriguing question.

The basic answer is that no one knows and that it depends on details, but let me elaborate.

An important perspective is that the inside of a black hole is very unlike a star. Stars, including white dwarfs with degenerate electrons and neutron stars with degenerate neutrons, are in hydrostatic equilibrium; the outward pressure gradients balance the inward pull of gravity. The essence of the space within the event horizon of a black hole is that everything is collapsing inward. In that case, the level of degeneracy of electrons or neutrons depends on how the black hole formed, from a collapsing star or otherwise, and, if a star, what kind of star. Another key point is whether the matter is degenerate or not, gravity will overwhelm any pressure, so any bit of matter must rush headlong into the “singularity,” in a short time – a fraction of a second for a normal stellar-mass case, somewhat longer for a supermassive black hole. With that perspective as background, the matter inside a black hole, in the brief interval after it passes the event horizon and before it hits the singularity, is likely to have some temperature and may or may not be degenerate. I cannot think of a reason that it should, at a given moment, be at absolute zero.

This perspective also addresses the second part of your question. Whatever density matter has as it crosses the event horizon, that density will rapidly increase as the matter plummets inward. Formally in Einstein’s General Relativity, the density goes to infinity at the singularity. We now understand that cannot be true. That condition of zero size and infinite density contradicts the quantum theory. We will not know what truly happens at the “singularity” until we have a complete theory of quantum gravity, string theory or something else.