Not necessarily. This may be more a matter of terminology than physics, but it is perfectly possible to have a system of particles that have non-zero kinetic energy, but which do not interact except that they all move in the potential of their combined gravitational field. This is the kind of system to which the formulation of the virial theorem that you gave (where the kinetic energy is minus one-half the potential energy) applies; the key is, as I said before, that if the only force acting is gravity, all the particles move on geodesics. But a fluid with a non-zero pressure is not this type of system; there are other forces than gravity present (the internal forces that cause the non-zero pressure), and they change the relationship between kinetic and potential energy.
For example, consider an "average" particle in the Earth's atmosphere, compared with a small test object in a free-fall orbit about the Earth at the same altitude. (We'll ignore the fact that an object in orbit inside the atmosphere would experience drag and would not stay in orbit; if you like, we can consider the second particle to be in orbit about a "twin Earth" that has no atmosphere but is otherwise identical to Earth.) The object in the free-fall orbit obeys the virial theorem in the simple form you stated it: its kinetic energy is minus one-half its potential energy. But the particle in the atmosphere does not; its kinetic energy is much *less* than minus one-half its potential energy, because its potential energy is the same (the altitude is the same), but its kinetic energy is just the temperature of the atmosphere in energy units, which is much smaller than the equivalent "temperature" of a particle in orbit. Put another way, the average velocity of a particle in the atmosphere is much *less* than the orbital velocity at the same altitude. So the virial theorem in the simple form you gave does not apply to a fluid with non-zero pressure.
The mass of the object *is* the total energy of the system; at least, it is if mass is defined in a consistent way (as the externally measured mass, obtained by putting objects in orbit about the body at a large distance, measuring the distance and the orbital period, and applying Kepler's Third Law).
As I understand it, the intent of the calculations is to study the static equilibrium states as a function of total energy (which is the same as the externally measured mass, see above), without specifying exactly *how* the system goes from one static equilibrium state at a given total energy, to another at a slightly different total energy. The key is that the total energy is specified "at infinity" (which in practice means "as measured far away", using the procedure I described above). I don't think any specific assumptions are made about how the total energy of the system is partitioned internally--how much of it is rest mass of the parts, how much is internal kinetic energy, i.e., temperature, and so on.
I think I agree with this, but I would have to do more digging into the details of how degeneracy pressure works to be sure. For the purposes of this discussion I'm fine with taking the above as correct.
As I noted above, I don't think the static equilibrium models go into this level of detail about how the system's total energy is partitioned internally. I understand what you're saying, but remember that, in relativity, energy and mass are different forms of the same thing; the energy that is added by compression, when the object is composed of degenerate matter, could just as well be taken up by nuclear reactions inside the object that effectively "store" that energy as the rest mass of newly created particles. That's basically what happens with neutron stars: the protons and electrons from atoms are forced by pressure to combine into neutrons, and the rest mass of a neutron is *more* than the rest mass of a proton and electron combined, so effectively the energy from the compression is being stored as rest mass. All this can happen without changing the temperature at all. But, as I said, I don't think the models that are predicting the overall equilibrium states go to this level of detail; they just adopt an overall equation of state for the matter making up the object.