Also, I've heard that one of the reasons that Charlemagne was crowned Roman Emperor was because the Papacy was theoretically/legally still a vassal to the Byzantines at that time, so naming Charlemagne Emperor was a way to free itself from Constantinople. If so, were Byzantine coins used in Rome during that period?
On papal history, see here.
Yes, popes minted coins. Lots of people did; minting a coin was a very powerful way to show that you were a sovereign ruler in your own right. However, this did not create a localized currency region like we have today. People used what they had, since everything was silver (in the West) or gold (in the East) anyway.