Yes of course! That is what motivates much of the study of neutron stars.
Tackling your first point, yes the interior temperatures of recently collapsed neutron stars are very high - perhaps $10^{11}$ K, but cool rapidly (in seconds) through the emission of thermal neutrinos to "cold" configurations where the Fermi energies are much greater than $kT$. There is plenty of theoretical interest in how matter behaves in these circumstances and how the neutrinos interact with other matter, since it controls the physics of supernovae.
The physics of neutron stars is reasonably well understood up to the nuclear saturation density at $3\times 10^{17}$ kg/m$^3$, but a typical neutron star probably has a density several times to an order of magnitude larger than this. The higher Fermi energies at these increased densities could well lead to exotic possibilities such as the creation of massive hadrons like hyperons or to the production of pions or kaons which then form a boson condensate. Alternatively, quarks may attain asymptotic freedom at high densities leading to a quark-gluon plasma or neutron stars that are entirely made of (strange) quark matter. Another alternative is that neutrons form some sort of solid core, held in a lattice by the strong nuclear force.
I suppose what you are really asking is when a neutron star exceeds the Tolman-Oppenheimer-Volkhoff limit and collapses, does the density increase even more to the extent that even more exotic (non-equilibrium) physics becomes possible? One thing to note here is that the radius of neutron star when it collapses is probably only a factor $\sim 1.5$ larger than its Schwarzschild radius, so the average density is only going to increase by a factor of a few before it departs the practically observable universe (I am not going to speculate on what cannot be observed). Given the current uncertainties in the behaviour of matter beyond the nuclear saturation density, I suspect that all the candidate physics considered for neutron cores is also relevant for collapsing neutron stars.
EDIT: Thus to answer your edit: At the highest densities inside neutron stars and at the highest densities achieved by material collapsing to a black hole (perhaps a few $10^{18}$ kg/m$^3$, the Fermi (kinetic) energy of the neutrons becomes large enough (a few hundred MeV $\sim$ the QCD scale) (or equivalently, the separation between neutrons becomes small enough $<10^{-15}$ m) that quarks may attain asymptotic freedom and quark matter is formed. The Fermi energies never get anywhere near the electroweak scale of 246 GeV.
