About how the Holy Britannian Empire is organized, I have a theory: it's actually based on the Holy Roman Empire in our own world. Look at the parallels. First off, although it was called the Holy Roman Empire, it was neither Roman nor based in Rome like how Britannia is (presumably) based in North America. Secondly, the States in the Britannian Empire may be like the States in the old Holy Roman Empire: semi-independent States. In the Holy Roman Empire, the Emperors were elected by Prince-electors. If Britannia works the same way, Charles di Britannia could be a power hungry Emperor-elect who has used his Geass powers to take complete control of the Empire. This would explain why Lelouche has no Uncles, Aunts or even Grandparents.
tl;dr Holy Britannian Empire = Holy Roman Empire