How is it that earth's inner core is still solid?
The melting point of the vast majority of materials increases as more pressure is applied. At https://www.researchgate.net/figure/312054885_fig1_Figure-231-Experimental-pressure-temperature-phase-diagram-of-pure-iron-a is a phase diagram showing how this variation occurs for pure iron:


From https://physics.stackexchange.com/questions/184032/what-is-the-pressure-at-the-center-of-the-earth-or-a-neutron-star the pressure at the centre of the Earth is 365GPa. At this pressure we can see from the graph that the melting point of pure iron is well over 5000k. Thus, given the experimental uncertainties and the that the core is not pure iron, it is easily conceivable that the core of the Earth is solid, or at least a large proportion of it is.