As with almost anything relating to this topic, this was a very long process that took place over centuries. I'll do my best to attempt to answer your question thoroughly.

When exactly Rome, the city became a less important factor in being Roman is difficult to pinpoint with an exact date. Initially, during the Roman Kingdom and early Republic being Roman meant...well being an inhabitant of Rome, the city. Only inhabitants of Rome could get Roman citizenship, and as a result be considered Roman. However as the Republic expanded and became increasingly multicultural this became a problem.

Over time certain concessions had to be granted to ethnic minorities, and very gradually more people got access to Roman citizenship. Citizenship came then to be used as a tool for gradual Romanization. Following the Social War Roman citizenship was expanded to all free born people in the Italic tribes, and by the late Republic Cisalpine Gaul was also granted citizenship. By the Principate certain Senatorial Provinces also could grant citizenship to the specific regions they encompassaed.

The year 212 A.D saw perhaps the most drastic and wide reaching change in Roman Citizenship, and as such the most drastic change in Romanity thus far. This year saw the Emperor Caracalla issue an extremely crucial edict granting citizenship to all non-slaves within the entire Empire. The importance of this edict cannot be understated, before this only Italians and certain colonies had full Roman citizenship, now every non-slave had it.

The effect this had on individual citizens was not immediate, but over time it resulted in a very substantial Romanization of the people within the Empire. Roman gradually came to replace the old identities of the people in the provinces, or at the very least became a secondary identity. In both Greece and Egypt, two provinces which had a strong native identity prior to the Romans took over, and had historically held onto this identity for centuries, Roman identity gradually replaced the native one. It was no longer uncommon for native Greeks and Egyptians to see themselves as and call themselves Romans, because that was what they were. Their nationality was Roman, and they had Roman citizenship, they were now given equal rights to the inhabitants of Rome, the city.

This is the first step in the city of Rome's decline, before this edict, to become Roman one would have to move to Italy or Rome in order to gradually aquire full citizenship. The people living in Rome, even the lower classes, de jure had more privileges and rights than those living in the provinces. Now this was no longer the case, and as such Rome lost some of it's political importance.

The importance of Rome was further weakened by the Crisis of the Third Century (c. 235-284), during this nearly 50 year period there were atleast 26 claimaints to the office of Emperor and the Empire itself saw an increase in foreign invasions and breakaway provinces.

Many of these Emperors were proclaimed by the army, and as such were proclaimed outside of Rome, ignoring the Senate's role to traditionally accept an Emperor. As the Empire was in an increasingly dangerous position militarily many Emperors also decided to not rule from Rome. Rome was strategically very poorly located in times of crisis. It was too far away from the borders of the Empire and was not in the vicinity of any major trade routes, worse still it was extremely difficult to defend. Rome was located on rather flat terrain and prior to the crisis had no effective city walls.

As a result Emperors increasingly spent their time in cities closer to the frontier, like Trier, Milan, Ravenna, Nicomedia or Sirmium. These cities were all located near some kind of frontier (Trier was close to the Rhine, Milan was close to the Alps, Nicomedia was close to Persia, Sirmium was close to the Danube etc.) and would allow the Emperor/army to quickly travel to the frontier in case of emergency.

As you can imagine this made Rome even more politically irrelevant, not only had it ceased to be as attractive for people in the provinces, but now the Emperor, the guy pretty much running things was nowhere to be seen. The Crisis of the Third Century ended with the reign of Diocletian, and Diocletian himself openly despised Rome. He saw the city as corrupt, dirty, overcrowded and dangerous, as such he avoided it as much as possible and is only recorded to have reluctantly visited the city once in his 20 year reign.

As you probably know once Constantine came to power he formally moved the de jure seat of the Empire from Rome to Byzantium (Milan had been the de facto capital in the west for a while), which he renamed Nova Roma (New Rome). The New Rome (Constantinople) had everything the old Rome had, it had a Senate, a Hippodrome, Temples, Churches, Aqueducts, Forums, the Imperial Palace etc. As a result the real seat of power and culture in the Empire was the New Rome, the Old Rome was an important and prestigious place, but it wasn't the capital or center anymore.

There was also a belief, popular around the time of Constantine that the borders of the city of Rome had expanded to the borders of the Empire. Rome was no longer just a city, Rome was a world, an idea.

It is therefore around this time, the 4th Century A.D that the city of Rome has finally fallen into complete irrelevancy. It was still a big city, but the Empire had plenty of those, aside from it's history Rome was not anymore important than Constantinople, or Antioch, or Milan, or Alexandria. I hope this has answered the first part of your question.

Now onto the second part of your question, as we know during the following century the Roman Empire slowly began fracturing politically. The Western half of the Empire effectively ceased to exist, but the Eastern half remaind relatively untouched. Why would they still consider themselves Roman if Rome the city, was no longer a part of their Empire?

Well as I hope I've explained, Rome the city wasn't as important. The way they saw it, they still owned Rome, because the borders of the city had grown to encompass the entire Empire. Everything that was a part of the Empire was a part of Rome, everyone that lived in it was Roman. The Old Rome may have been lost, but the New Rome was still alive and thriving. The Roman way of life, Roman laws, the Emperor, the Senate was all intact, they had no reason to think they were anything but Roman. They still paid their Roman taxes to the Roman governors who taxed them based on their Roman citizenship so the gold could be used by the Roman Emperor, just as it had always been.

Your last question I think is a little strange. You see there was no such thing as a Western Roman Empire and a Eastern Roman Empire, these are historiographical constructions. Nobody in 395 suddenly became a East Roman citizen and lived in a seperate state from a West Roman. The Empire was still one single Empire, it was just ruled by more than one Emperor out of necessity, as had been the case before. Sometimes these two Emperors cooperated, other times they didn't, sometimes one of them reunified the Empire, other times this was not the case. But do not fall into the trap of thinking these are two wholly seperate Empires, the division of 395 is no different from the Tetrarchy or even the Triumvirate during the Republic. The Empire is divided among factions to better administrate it, it's not partitioned between two new Empires. As such there was no date in which the "formation of the Eastern Roman Empire" took place. Someone ruled the Roman East seperate from the West on and off from the time of Caesar until 480.

So when did the Roman Empire end? When did it cease being truly Roman? Well frankly that depends entirely on what one subjectively believes it means to be Roman.

To the Romans themselves they were always Roman, all the way until 1453 and even after 1453, despite being deprived of an independent state. But let's throw some other subjective factors around just for fun.

If a continious Roman state apparatus is the defining factor for the Empire being Roman, then the end of the Empire would be in 1204, when the Empire was partitioned following the 4th Crusade.

If culture is the defining factor for the Empire being Roman then the end of the Empire would be in 1453. The culture may have been drastically different from the classical Empire, but cultures change all the time. They still saw themselves as Roman and alot of developments in their culture even in 1453 could be traced back to the Classical period.

If religion is the defining factor for the Empire being Roman then the end date would be 380, when Emperor Theodosius made Christianity the state religion.

If the language of the state is the defining factor for the Empire being Roman then it ended somewhere between the reign of Heraclius and Leo the Wise (c. 600s-800s A.D).

Tl;dr: It really depends entirely on how much emphasis one want to place on these individual factors.

Thanks for your thorough explanation.

Let me follow up then on some points you raised.

I read somewhere that Christianity could be defined as the single biggest reason why the Empire lasted as long as it did. By making the Roman identity inseparable from the faith, it kept the Empire together in a way that the varied pagan cults and blind loyalty to a national identity never would. Do you feel like the Empire could have found its cohesion had Milvian Bridge gone against Constantine, or Julian succeeded in stopping Christianization in its infant stages?

It’s funny that you raise Caracalla and his edict to grant citizenship to everybody in the Empire. I was just thinking about that recently. So, it seems like you’re saying that this edict, plus the extension of the borders of Rome to encompass the entire Empire, contributed the most to the idea that Rome had risen from being a mere city to being a concept, and to be Roman meant not that you were a citizen of the city of Rome, but that you were a citizen of the empire itself, which was in a way the city as well.

Do you feel that the repeated sacking of Rome had anything to do with its reduction in importance?

Also, if Rome as a city wasn’t as important anymore, then why did the Pope, as the Bishop of Rome, retain such importance for the Empire versus the Patriarch of Constantinople?