Why is Earth's inner core solid?
Earth's inner core is solid even though the temperature is so high because the pressure is also very high.  According to the Wikipedia article on the Earth's inner core, the temperature at the center is $5,700\ \text{K}$ and the pressure is estimated to be $330$ to $360\ \text{GPa}$ ($\sim3\cdot10^{6}\ \text{atm}$).  

The phase diagram shown below (taken from this paper) shows the liquid/solid transition, where fcc and hcp are two different crystalline forms of solid iron.  You can see clearly from the slope of the line going off toward the upper right that iron should be solid at this temperature and pressure.