Structure of Buckminsterfullerene
Have you ever seen a picture of graphite, to begin with? It shares the same feature, i.e., that each carbon is bonded to exactly 3 other carbons. You may call it "alternating double bonds", but this is not quite correct either, because in fact all bonds are equal. Think of it as a resonance structure, much like benzene. This is the reason for graphite conductivity, too.

The idea about that lone pair is plain wrong. Carbon has but 4 valence electrons; once 3 of them are spent for sigma bonds, how could it have a lone pair?

Pure buckminsterfullerene does not conduct electricity (strictly speaking, it is a semiconductor, not a conductor like any metal). Some of its derivatives do, but that's another story.