RE: iron core being Molten: Even though it is hotter than the surface of the Sun, the crystallized iron core of the Earth remains solid
Matter phases depend heavily on both temperature AND pressure. Under pressure, some things will refuse to melt. That is why some folks think that even gas giants and smaller, colder stars may well have solid cores.
The math specific to the theoretical derivation depends on too many factors. For that reason, I will not offer any detailed analysis. It is my OPINION (clearly stated as such) that some parts of the core may be molten since it is the spinning motion that creates our magnetic field and if everything was solid deep down there, we would not have any kind of spinning other than in exact lock-step with planetary day/night cycles. But the scientists who have studied this say we don't have that situation.