Uuh, that would be the Empire established by Constantine as well as for centuries of rule by the Holy Roman Empire whose allegiance was demanded NOT of and for Jesus Christ, but of and for the Pope/Emperor.
Btw, technical semantics aside, history notes that the "Vicar of Christ" has been anything but in most cases.
Then you are quite wrong
The Byzantine Empire soon enough forgot about Rome and it was influenced more by the Alexandrian, Antiochian and above all the Constantinople Patriarchs
as well as for centuries of rule by the Holy Roman Empire
Again wrong -- you have heard of the Guelphs and Ghibellines, right? That presence in Italy is enough to tell you that the Popes didn't "hold Dominion over the HRE" -- some may have tried, but beyond the ability to crown the HRE, they didn't have any pull to "hold dominion"
whose allegiance was for the Pope/Emperor -- again, completely wrong -- please do read some history -- the oath went ""I swear, by God and the Holy Bible, to righteously serve my Kaiser, Holy Roman Emperor and King of all Germans, until the day I die. I will defend the Holy Roman Empire and the people of it, with my life if necessary. I will also uphold and follow the laws of the Empire and serve the Reichstag with honour and dignity. May God send his divine punishment if I ever break my oaths to the rightful Kaiser and the Holy Roman Empire.""-- to the Emperor, not the Pope
So this disproves your earlier statement cmpletely