Well... Can we really call it a star if there is no fusion? IMHO, this makes a brown dwarf a type of planet.
It isn't really a planet either - its interior is fully convective, like a star's interior, so there is no stratification of composition with depth. Ten thousand miles down it's the same as ten feet. All planets show differentiation with depth. Even Jupiter, if you go down a few hundred miles, transitions from a gaseous atmosphere to an ocean of liquid metallic hydrogen.
I have a stupid question... what language does the name 'Earth' come from? Is that just an English word? I think we should call this planet Terra, officially. It would fit better with the Greco-Roman naming scheme for the rest of the solar system. Plus, it sounds better being referred to as 'Terrans'. 'Earthers' or 'Earthlings' doesn't have the same ring to it.
^ It's all a question of which language group you want to use. "Earth" and similar names are from the Germanic languages, "Terra" and similar names are from the Latin languages, and "Gaia" and similar names are from Greek. I can't honestly think of any good reason for one of those groups of languages having priority over another.
Physically impossible. A body of gas with four times the mass of Jupiter with or without a rocky core has to increase in density as you approach it's center. Do you mean chemical composition? Where are you getting that from?
Yeah chemical composition. Of course the density increases as you said. Even in a star this remains true, otherwise the core would not sustain fusion.
A star's interior is not fully convective. Only the analogue of the mantle is convective in a star (well, the photosphere also, but that's turbulent convective, and generally detached from the convective zone). In the analogue of the outer core, a star, or at least the sun, has a radiative zone. Nor is the chemical composition uniform in a star - the innermost part of the inner core is far richer in helium than the rest of the star, as gravity pulls the heaviest elements to the core (which is really too bad... if there was a way to reverse that, force all the helium to the surface, we could prolong the life of the sun by a good 20 billion years as a conservative estimate). In aging large stars, as you approach the center of the star you encounter shells of progressively heavier elements.