You're right that gravity doesn't increase all of the way to the core -- in the Earth, it increases down to roughly the middle of the mantle, and then it decreases down to nothing at the very center. So the settling of heavier elements "should be" most pronounced in the top half, and less pronounced, or even insignificant, in the bottom half. Just remember that as the gravity diminishes with proximity to the core, so does the centrifugal force, which varies with the radius. So it isn't going to be light elements in the core, and heavy elements at the surface, due to centrifugal separation. At the rotation rate of the Earth, gravity is always the more powerful force.
If something was stirring or pumping the liquid at any of these levels (such as electrostatic discharges, which might prefer one element over another), that might be a far bigger factor than simple mass separation due to gravity.
As concerns the geomagnetic field, if it's caused by ferromagnetism (which I don't think it is), the core would have to be iron, nickel, cobalt, etc. But if the field is caused by rotating charged particles, the elements could be anything, since all elements can get ionized under sufficient pressure.
Bottom line: I'm no help on this one.

If we could accurately estimate the overall density of the Earth, we could set limits on how much of the heavy elements versus light elements were present, knowing that somehow, it all has to average out to the estimated density. But then again, how to estimate density is the central topic of this thread.
