Nope. What happens is that there are more energy states available, and so the effect of having the limited number of energy states disappears.
If you can have fermions at relativistic states, then degeneracy pressure should disappear. Now figuring out how to set up that sort of experiment in the lab is something I'll leave for other people to do.
Not quite. What happens is that the energy levels change so that fermi and boltzman distributions converge to something that is different than non-relativistic gases.