What was relationship between the HRE and the Byzantine Empire?
How did the two nations deal with two emperors claiming to be Roman?
Exactly how much of an impact did the fall of Rome have on Europe? Did any nations attempt to replicate the Romans?
Thank you!
There's always much more to be said about this - the Eastern Roman Empire and the Holy Roman Empire had a very complicated relationship that was sometimes good and sometimes bad over the 5 or 6 centuries that they both existed. But I have some previous answers about two specific incidents that might be helpful:
What was the Eastern Roman reaction to Charlemagne being crowned Roman Emperor by the pope in 800?