It really depends on which author you are talking about, but yes, to a great extent they did. This does not mean that they were blind to the relative political situation of Rome and her neighbors having changed quite a bit between 100 and 400 CE, but Rome was still central, even if this was often phrased in apocalyptic terms. For example, Jerome's Letter 127.12 reads thus:

My voice sticks in my throat; and, as I dictate, sobs choke my utterance. The City which had taken the whole world was itself taken; nay more famine was beforehand with the sword and but few citizens were left to be made captives. In their frenzy the starving people had recourse to hideous food; and tore each other limb from limb that they might have flesh to eat. Even the mother did not spare the babe at her breast. In the night was Moab taken, in the night did her wall fall down. Isaiah 15:1 O God, the heathen have come into your inheritance; your holy temple have they defiled; they have made Jerusalem an orchard. The dead bodies of your servants have they given to be meat unto the fowls of the heaven, the flesh of your saints unto the beasts of the earth. Their blood have they shed like water round about Jerusalem; and there was none to bury them."

You can see the way the traditional position of Rome at the center of the world had been absorbed into Christian thought, making it ripe for comparison with the many gloomy passages in the Old Testament regarding Jerusalem. At this period of history, Catholic Christianity was, with exceptions, confined to the borders of the Empire. The Saharan frontier was populated mainly by pagans, the German frontier was largely Arrian, and the Persians to the east were Zoroastrian. So within a Christian context there was a real view that the Empire was the preserver and protector of the faith.

Of course this is a view that Augustine attacked in City of God from anti-materialist and anti-imperialist perspectives, which goes to show how popular it was at the time.

But there was also a more secular perspective on this. Ammianus Marcellinus is often considered the last great Latin historian, and he was indifferent if not outright hostile to Christianity. He begins Book XXI, the last book in his history in which he discusses the battle of Adrianople, thus:

The following circumstances were the original cause of all the destruction and various calamities which the fury of Mars roused up, throwing everything into confusion by his usual ruinous violence: the people called Huns,

etc etc, which is a rather gloomy way to start the close of a book.

So the Roman Empire was still very much thought of as the center of the world, and the phrase orbis terrarum (circle of all lands) was still used to describe the empire.

This is a huge topic so my answer will definitely not be able to cover all of it, further clarifications and additions to my answer are very welcome! It is also difficult to think of specific sources since the idea of Roman superiority, at least for me, is so ingrained in the stuff I've read that it's not something I've thought about too much.

I think the short answer here is that yes, the Romans still thought themselves to be better than everyone else, with references to the glorious past being omnipresent in official texts. However, what the sources say don't necessarily represent the mindset of the 'average' Roman, and the repetition of certain themes may just be rhetorical tropes rather than anything else, so be aware that this entire concept may just be a construct, one that a lot of people didn't buy into. However, I think it would be sensible to say that Christianity did not transform society utterly and likewise the detente with Sassanids from the fifth century onwards only required a slight alteration to the conventional worldview, so much of the stuff you identified remained the same. As for the 'barbarian's soldiers, I'd say that most were no less loyal to the idea of the Roman Empire than 'proper' Romans, if only because of the fact that the empire was useful and often superior to any neighbouring military powers, so why would they think otherwise? Cicero's rhetoric was still being taught, Roman armies were still winning victories and emperors continued to maintain Roman traditions, to name just a few examples that demonstrate the continuity they still had with the past.

Those who were not aligned with Rome were just as unlucky as before, as they were regularly condemned as barbarians and savages by most authors between the fourth and the seventh centuries, though I will also point out a few interesting things in the sixth century, as the vacuum left by the disappearance of the Western Roman Empire led to some 'barbarian' states to adopt aspects of imperial authority - the prestige of Rome was still great, but the concept itself was flexible and so was able to be used by many different sorts of people. I think the best way to illustrate this is through a few relevant quotations, especially as I don't have the theoretical knowledge to talk about this topic at length.

Our best source for this is probably imperial propaganda, since the image the emperors wanted to propagate tells us a lot about what the ideal situation should be, regardless of how many problems they faced in reality. Throughout this period, they trumpeted the power of the emperor and how he crushed his barbarous foes. I can dig up some specific examples if you want, but court panegyrics often followed the same themes and so can get pretty repetitive. A more fun example would be Ammianus Marcellinus, a pagan historian writing in the late fourth century and thus after the notable Roman defeat at Adrianople in 378, but he repeated the same ideals past Romans had promoted:

At the time when Rome first began to rise into a position of world-wide splendour, in order that she might grow to a towering stature, Virtue and Fortune, ordinarily at variance, formed a pact of eternal peace; for if either one of them had failed her, Rome had not come to complete supremacy. Her people, from the very cradle to the end of their childhood, a period of about three hundred years, carried on wars about her walls. Then, entering adult life, after many toilsome wars, they crossed the Alps and the sea. Grown to youth and manhood, from every region which the vast globe includes, they brought back laurels and triumphs. And now, declining into old age, and often owing victory to its name alone, it has come to a quieter period of life. Thus the venerable city, after humbling the proud necks of savage nations, and making laws, the everlasting foundations and moorings of liberty, like a thrifty parent, wise and wealthy, has entrusted the management of her inheritance to the Caesars, as to her children. And although for some time the tribes have been inactive and the centuries at peace, and there are no contests for votes but the tranquillity of Numa's time has returned, yet throughout all regions and parts of the earth she is accepted as mistress and queen; everywhere the white hair of the senators and their authority are revered and the name of the Roman people is respected and honoured.

Ammianus then embarked on a famous rant about how the senatorial elite had declined in virtue and that the Rome he lived in was an inferior city compared to the glories of the past ("[...] these and similar things prevent anything memorable or serious from being done in Rome"), but it is nonetheless clear that for him, the Roman Empire was still the heart of the world, regardless of how weak it was in reality. Ancient history was happily recounted and old heroes framed as relevant models, so very little had actually changed, especially as education for the elite generally followed the same model regardless of religion. Ammianus was a pagan operating in an increasingly Christian world, yet his worldview remained largely the same.

The same was true when Rome faced another crisis, the Sack of Rome in 410 by the Visigoths. People did panic, but again it was framed through the perspective that Rome was the most important city in the world (which made the Sack all the more shocking), as seen here in the reaction of St Jerome:

Whilst these things were happening in Jebus a dreadful rumour came from the West. Rome had been besieged and its citizens had been forced to buy their lives with gold. Then thus despoiled they had been besieged again so as to lose not their substance only but their lives. My voice sticks in my throat; and, as I dictate, sobs choke my utterance. The City which had taken the whole world was itself taken.

People were of course aware that the empire had faced massive defeats in the past, so this provided a way out for people trying to explain recent failures. Bishop Ambrose of Milan in an earlier letter for example used the prisoner-emperor Valerian and the Crisis of the Third Century to show that the Roman Empire had survived previous crises and that imperial failures had nothing to do with what beliefs the emperor had. The same goes here, most evidently in the retort penned by Orosius, who was trying to argue for the same sort of thing. For him, although the empire did suffer a major blow, it was still able to recover:

Although this deed [Sack of Rome] is of recent memory, if anyone were to see the great numbers of Rome’s population and listen to them, he would think, as they themselves say, that ‘nothing had happened’, unless he were to learn of it by chance from the few ruins which still remain from the fire.

I'll have to skip the rest of the fifth century, as I'm less familiar with it, and move on to the sixth, where the same themes were repeated. The preface to the Justinianic Code, a key part of the reforms brought about by Emperor Justinian, had this to say:

The maintenance of the integrity of the government depends upon two things, namely, the force of arms and the observance of the laws: and, for this reason, the fortunate race of the Romans obtained power and precedence over all other nations in former times, and will do so forever, if God should be propitious.

Christianity only added to the existing rhetoric of power promoted by the government and did not change fundamentally how the Romans saw themselves. This is best seen in sources from outside the empire, particularly, from the west, since they tried their best to match their Roman heritage even though they lived in a post-Roman world. The empire meant a great deal to them and acknowledging that was a fact of life, even though of course political realities often meant that they sometimes had to compete with Constantinople. Theodoric the Great was for example described thus in the Anonymus Valesianus:

He so governed two races at the same time, Romans and Goths, that although he himself was of the Arian sect, he nevertheless made no assault on the Catholic religion; he gave games in the circus and the amphitheatre, so that even by the Romans he was called a Trajan or a Valentinian, whose times he took as a model.

Likewise by the Greek historian Procopius:

And although in name Theodoric was a usurper, yet in fact he was as truly an emperor as any who have distinguished themselves in this office from the beginning; and love for him among both Goths and Italians grew to be great.

Indeed, it is possible to argue that Theodoric was seen by his subjects as a new emperor of the west, which in itself testifies to the flexibility of the idea of empire, as well as its overpowering dominance over the discourse of this period. Rome was still superior, but that did not stop 'barbarians' from being full members of the Roman world.

This is also illustrated through the empire's relationship with Sassanid Persia. The two empires remained at peace for most of the fifth century and although the sixth century was a period of increasing conflict (see my comment here), co-operation was also common. There was a complicated, almost ritualised form of contact between the two empires, one that culminated in the rhetorical integration of the Persian Empire into the imperial mindset by the beginning of the seventh century, which is most clearly seen in this description of the two empires by the historian Theophylact Simocatta, which also neatly sums up my arguments here:

God effected that the whole world should be illumined from the very beginning by two eyes, namely by the most powerful kingdom of the Romans and by the most prudent scepter of the Persian State. For by these greatest powers the disobedient and bellicose tribes are winnowed and man’s course is continually regulated and guided.