Sorry for lots of questions, but I cant seem to find the answers easily and I'm curious as to whether theres any sources that give even just hints at what the answers could be...
If the Eastern Roman Empire (or Byzantine Empire) never referred to themselves as "Eastern" or "Byzantine Empire" while they were in existence, but merely the "Roman Empire" - how did the Emperors of Constantinople rate himself vs the Emperors of Rome, did they work together to run the Empire as if it was one? Was this like a political division, or an actual physical division? The two Empires adopted different religions (Latin Christianity - Pope, Greek Christianity - Patriach), did this create friction between them? Did they ever go to war with each other?
Also, how did the Eastern Roman Empire deal with identifying as Romans while the Empire didn't hold durisdiction over Rome? Surely that would cause a kind of national identity crisis from claiming to be Romans while another Empire, also claiming to be Romans, actually holds Rome?
Lastly, during the time period that the ERE owned Rome the city itself, what was Rome's role in the Empire? I assume it wasn't made capital as my own research might've come across that, but was it held as simply a province, or was it given any special note within the ERE?
The Empire was first divided by Diocletian at the end of the 3rd century AD. He ruled the eastern half, and delegated the rule of the western half to a subordinate. Each half was further divided into two four a total of four (which leads this period of government the name 'tetrarchy'). While this initial split set an important precedent for the later splitting of East and western Europe, it did not last and the Empire eventually went back to a single unit for a time.
Because Diocletian chose to rule from the east rather than the west and instead chose a subordinate to rule from Rome, I would guess that the later Byzantine emperors believed that they were rulers of the eastern Roman Empire, rather than Roman citizens who ruled an Eastern Empire. In other words, by the Byzantine period the idea of Roman culture referred to the culture in the Roman Empire rather than in the city of Rome.
Also, the western Empire had fallen long before the great schism between Catholicism and orthodoxy; when the western half fell what would become catholicism was practiced throughout the Empire.
As already mentioned, the first division under Diocletian was essentially administrative - it turned the Empire into two halves for the purposes of providing each half with a competent (in theory) ruler to manage, defend, and control it. After Diocletian this worked sometimes well, sometimes less well. The brothers Valentinian and Valens seemed to do okay with it. Constantine's 3 sons couldn't deal with a divided empire very well and almost immediately resorted to civil war.
The Catholic church was not ultimately divided by schism until 1054. There were periods of schism between East and West before that (the Acacian schism, Photinian schism) for example, but it is incorrect to say that there were two different religions. Certainly up until the fall of the Western Empire, say 476, the church would have seen itself as one.
Rome declined in importance during this period of the Late Empire. The administrative centre for the West under Diocletian became Milan, not Rome. The only claimant for Western emperor who was based in Rome was Maxentius, in 306. Final division of the empire really came after the death of Theodosius in 395, with Honorius becoming (puppet) ruler in the west, first at Milan, then at Ravenna. Rome never regained its importance as the capital.
The loss of Carthage to the Vandals in 439 cut off significant grain supplies from North Africa, and grain from Egypt was by this stage mainly supplying the eastern Empire and Constantinople, so lack of grain imports was also a factor in the population decline in Rome.