Short answer: 1) Yes and no; 2) Yes, there is a supercritical fluid, of hydrogen.
It's fairly hot deep down in Jupiter; estimates range from 10,000 K to 24,000 K. You would think that anything in the core would be liquefied, and you could be right. Many models predict that Jupiter's core is rocky, but others predict that it is liquid. Still others say that it is "liquefying" - i.e. parts are liquid and parts are solid. We don't know for sure which are right.
In the early solar system, Jupiter would indeed have to have had a large core of rock (and potentially ice). It would have accreted gases, and eventually the planet we know today would have formed, with thick cloud layers complete with hydrogen, ammonia, and lots of other elements. But after Jupiter accumulated its mantle, there would have been a lot of pressure on the original core. It seems like much of the matter inside it would have been liquefied and carried away ("redistributed") into other parts of the planet. Still, some might have remained.
Surrounding the core, there is a layer of liquid hydrogen. Yep, liquid hydrogen, just as you said in your question. Temperatures and pressures are so high that hydrogen has passed its critical temperature (33 K) and become a supercritical fluid.
So, in summary, we don't know for sure the state of Jupiter's core is, but parts or all of it may be liquefied, and we also think that there is indeed a layer of hydrogen as a supercritical fluid surrounding the core. The same may also be said for other gas giants, although, still, we don't know for sure.
I hope this helps.
http://arxiv.org/pdf/1111.6309v1.pdf (You might find this particularly interesting, as it considers gas-giant exoplanets)