jtb,
Yep, heat is energy

(actually, at some level, everything is energy [E=mc^2]).
The earth gets energy from the sun as well (and it's important), but without the warming effect of the core, the climate on the earth would probably be much like that on the moon, either really hot, in the sun, or really cold, in the dark.
IIRC, it's generally considered that the combination of inner earth temperature, nice orbital position (right amount of sun), and enough mass to retain a decent atmosphere is what makes earth as habitable as it is.
IPlural, what you're talking about is the 'snowball earth', a state that did exist quite some time ago (couple billion years, iirc). The increase in diameter from ice was negligable, but one of the things that allowed the ice to melt at all was volcanic action (without it, the ice reflected a lot of the energy of the sun back into space, with not enough left to melt the ice by itself).
Lightning isn't plasma, although it generally creates some in the air as it ionizes it. The plasma that most people are familiar with is a common flame.
The core of the earth is solid, not because it's a special state of matter, but because at the extremely high pressures at the core, the metal is pressure frozen, despite being at such a high temperature.
Also, the temperature of the inner earth drops as it gets closer to the crust, the outer core is cooler than the inner core, and the mantle is cooler than the outer core.
As for the crust, 30 miles is pretty much it (i think the range is from about 4 miles at the thinnest to about 37 miles at the thickest parts).