YES. The inner core is a solid sphere of iron +10 % Nickel, plus a minor amountt of other elements. Originally the inner core was liquid, but cooling of the Earth over the past 4.5 billion years has caused enough heat to be removed such that the Temperature of the inner core has fallen below the melting point of Iron alloy at the pressure of the inner core, which is very, very high.

Yes, the inner core is solid. It is mostly made of a nickel-iron alloy. Although it is very hot in the center of the earth, the metal doesn't melt that easily.

Good question! The brief answer is: part of the core is solid, and part is molten (liquid rock). The outer core starts at a depth of about 2900 km (1800 miles) beneath the surface of the earth, and is thought to be molten iron, nickel, and other heavy elements. The inner core starts at a depth of about 5150 km beneath the surface, and is though to be solid. The composition of the inner and outer core are though to be similar, but the pressure is so high in the inner core, that the iron and nickel are stable as a solid, and not as a liquid! It is probably still very hot in the inner core (just as hot as the outer core), but it turns out that really high pressures can prevent things from melting.