Is it possible that the centre of the earth could be a super dense conductive carbon?

No. Seismology indicates the Earth has a solid inner core surrounded by a liquid outer core and both are very dense, too dense to be carbon. They're probably mostly iron, alloyed with some nickel and a little sulphur, because out of the elements that have the right density under the conditions in Earth's core iron is by far the most common in the universe.

No... observations of seismic waves, and simple physics suggest that the center of the Earth is a heavy, dense ball of iron and nickel. Carbon, being much lighter, would work it's way toward the surface during differentiation.

ope.

We already KNOW that it is molten iron.

We KNOW this because only IRON can generate the earth's magnetic field. Carbon can;t do that.

No. It's iron - magnetic field remember?

Carbon is light, so will be nearer the surface. Iron is heavy and sinks to the centre. Simple.

Nah...less on the dense, so a thin layer of carbon I think resides in the center of Earth, assisting the lava and helping out with the gravity.