These are close but I see room for nuance. When the empire split there were two emperors for a long while; after about 480 there was only the Eastern Emperor left and so the East got later to be called the Byzantine empire (after its capital Byzantium/Constantinople) rather than the Roman as it had always called itself. From the fall of Rome, for 300 years you had chaos and tribals in the west and for 1000 years you had stability in the east (largely due to the Byzantine gold standard, the best and stablest in world history). There was no political east-west divide after 480 because there was no Western Roman Empire left. Rome was either pagan/Arian, or run wholly by the east.

Did Eastern Emperors try to divide the church? They probably contributed as everyone did. The Pentarchy was nominally led by Rome and Constantinople, which were recognized as first and second among equals. Over time the assertions to Rome's privilege as first diverged so far, partly due to the communication divide, that in the 11th century issues were made over the right to add "filioque" to the creed and the right to excommunicate (both sides excommunicated each other, which was annulled by both in the 20th century). This was recognized as the great schism but both sides thrived afterward: a real Judah-Israel issue.

Until the 11th century, all the Pentarchy heirs were the one true catholic orthodox church of Jesus Christ founded on Peter. The Orthodox do not descend straight from Rome or Peter but from the "equal" grant given to Andrew and then Byzantium, just as the other original cities descended from other apostles. To speak of the origins of Catholics, Orthodox, or Protestants is always to speak of the plenary grant from Jesus to his apostles, and only to narrow that grant due to later events: all three camps should rightly lay claim to the entire church up to 1054. That catholic orthodox millennium is indeed the originator of all subsequently defined churches.

I don't know which Constantine you mean. I and II were over the whole empire, there were two Constantine III's, one for each side, and there were IV-XI in the east along with a few who took the name without a number. If anything Justinian I (Eastern Emperor 527-565) did more to establish Byzantine Christianity and control over western lands than most other eastern emperors, long before eastern Constantine III.

You might say instead the HRE was the answer by the Roman bishop (the pope) to the Eastern Roman Empire. But the HRE was really the "First Reich" of the German people, who went through very many somersaults to retain its alleged continuity after 1806. The Byzantines lost continuity to the Ottomans in 1453; some trace their elites to Russia where they contributed to the czar dynasty being set up. So as I said the political structures come and go; meanwhile the covenant people's structures remain under the one banner of Christ.