The author says that
In general this is a very difficult problem as the fermions cannot be treated classically. The coupled fermion-Maxwell-gravity system becomes tractable in a limit in which the fermions may be treated locally in the bulk as an ideal fluid of zero temperature charged free fermions.
in ArXiv:1008.2828
I cannot understand these sentences.
Due to the Pauli principle a finite density of zero-temperature fermions behave different from a classical gas (where all particles are at rest at zero temperature), they will have finite kinetic energy and exert a pressure.
No, very far from that! One first solve for all states in a non-relativistic (or relativistic) particle-in-box problem (or even simpler a particle-on-3-torus problem). Next one says that each such state can be occupied by only two (or more generally by a finite even number of) fermions. Then one calculates the total energy of this configuration, and how this energy depends on the number of fermions and the available volume.
You should study a little about ideal fermi gases to get a better grip on what I tried to tell you above. It is really an elementary application of first-course quantum mechanics. But an extremely important paradigm in solid state physics!
Because Fermi-Dirac statistics, like Bose-Einstein statistics, aren't classical notions, that's the reason. They describe quantum properties of matter. In the classical limit both become Boltzmann statistics.
Because then it's the properties of the charged, perfect, fluid that are used as an effective description, i.e. its equation of state. The individual constituants aren't probed. That's (part of) the content of section 2.
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