Was the Empire that Charlemagne ruled really the old Roman Empire - Was he *really* a Roman Emperor, or was he the ruler of a completely new political entity?
I would say it was a completely new political entity. Being crowned "Roman Emperor" by the Pope was a reach of desperation by the Catholic Church to keep themselves at the seat of power in Europe. By crowning Charlemagne it gave the image that the Church was bestowing upon him power, even though he built his empire without the help of the Church.
I believe that Charlemagne's title was all about propaganda. He wanted to associate himself with the great emperors of Rome and gain status by association. However geographically (as mentioned above) he was ruling a different empire than that of the Romans and was ruling them in an entirely different way. Charlemagne used the title to his advantage, but it doesn't make him the next in the line of Roman Emperors.
I'd say it was a new entity. He was not ruling the same territory that the Romans ruled. He was not using the same sorts of political institutions that the Romans used. Yes, it was the biggest European empire since the Romans, but that doesn't make it a Roman empire.