I will leave your follow up questions regarding what would become the patriarchates of the Eastern Orthodox Churches to others, but can happily tackle the status of the pope prior to the 11th century. The below is largely adapted from a previous answer, edited for clarity and some updated points of emphasis.

By the fifth century, the bishop in Rome held the was widely know by the informal title of papa, from which we get the modern title: pope. The title was not unique to the Roman see, and was commonly used to refer to a senior bishop. The title would, however, become unique to the apostolic see by the eighth century. The papa of Rome was one of five patriarchs (Rome, Constantinople, Antioch, Alexandria, Jerusalem) that placed them atop a largely decentralized ecclesiastical hierarchy. Even among these, Rome was singled out as special—it was after all the resting place of not one, but two apostles, on top of countless martyrs. The legacy of the city as the old capital of the Empire was also hard to shake. And the eastern patriarchs were willing to defer to Rome as the representative of the western bishoprics, which made since practically-- in the church politics and Christological debates of the fifth century, the pope was frequently brought in as a counterweight. In other words, the pope was 'special' insomuch as they were a recognized heir to a prestigious apostolic legacy and could act as a tiebreaker of sorts. But it's important to note that they were not 'above' the others in a meaningful sense. Even among the bishoprics of the West the pope lacked any formal jurisdictional or theological authority.

The papacy of the earliest medieval period was largely oriented towards the East and Constantinople, and as a result was often caught up in the theological disputes and politics of the Eastern Roman Empire-- central and southern Italy were part of the Empire for much of the period. Until the papacy of Zachary in 741, the election of a new pope required confirmation from the Emperor. As mentioned above, as the only Latin patriarch, the pope represented the West (in theory) in eastern affairs, but in terms of real, enforceable authority, the pope exercised little to no control over the western bishops and certainly none over its eastern counterparts. Like other patriarchs, the pope had a number of civic responsibilities. The sees of Ravenna and Rome paid the imperial armies in Italy and Rome fed the poor of the city from its estates. The acquisition of civic and administrative duties was not itself unusual, but among the metropolitan bishops of the West, Rome was particularly wealthy; the patrimony of St. Peter during the papacy of Gregory the Great numbered over 400 estates, many of which were located in Sicily.

Despite its eastern orientation and the increasingly fragmented condition of the western churches, the apostolic see still held a position of special deference in the West. During the fifth century, bishops from Gaul would occasionally petition Rome (addressing him as papa) or, as in the case of some African bishops, seek intervention in local conciliar affairs. This, however, was not regular, and more often than not the bishops involved were in trouble back home. As the western half of the empire continued to disintegrate, regional bishops became increasingly focused on the affairs of the various successor kingdoms to which they belonged. With the conversion of the Visigothic nobility from the 'Arian' heresy, the ecclesiastical life of that kingdom became increasingly centered on Toledo. The bishop of Arles in Gaul, theoretically and at times functioned as the papal representative in the region, but Rome lacked real jurisdictional authority over the bishoprics of the West.

Again, it's important to emphasize the slow and incremental changes in papal prerogatives in the West. Fifth century popes like Leo and Gelasius both emphasized the importance of their spiritual authority, but it's hard to gauge the extent of their vision for the role of their office; practically, they were unable to impose their will on the Latin bishops. Gregory the Great at the close of the sixth century certainly grew the prestige of the office, and while there was indeed an expansion of the papal bureaucracy over the course of the sixth and seventh centuries, the authority of the popes rarely extended beyond Rome, the surrounding territories and at times in Northern Italy.

The eighth century marks an important inflection point, when the papacy began to look north to the Carolingian Franks rather than East to Constantinople. The Franks increasingly looked to Rome in matters of the liturgy and instruction. Even still, the real authority of the papacy was limited--but it was growing. Beginning with Louis the Pious, the pope would take a more prominent role in the anointing of emperors, a reversal of the days when the Eastern emperor confirmed the pope’s position (though the Frankish emperors were still usually notified of the election of a new pope). Over the course of the ninth century, archbishops in greater numbers sought the conferral of the pallium, the ceremonial woolen vestment symbolizing the authority of their office, from the pope. It was an old practice, but the increased frequency speaks to the extent to which the ecclesiastical leaders of Francia were beginning to look towards Rome for legitimacy, even if the pope as of yet was unable to dictate who was elected.

The tenth century saw the continuation of several trends established in the preceding centuries and the papal offices were quite busy, even when the popes themselves were rather lackluster. Charters connecting the papacy to various corners of the Latin West are a testament to the increasingly centrality of Rome. Monasteries in particular sought the protection and patronage of the apostolic see—and it is out of these relationships that we witness the birth of the reform movements that would characterize the papacies of the eleventh century. Otherwise the eleventh century would look back on the tenth with not a little disdain. Papal election had always been a political affair, but the tenth century is notorious for the contests between leading Roman families, West Frankish rulers, and German kings—though the condemnation of the period overlooks continuity and consistency with earlier papal models. It is also at the end of tenth century that we witness the first ‘official’ papal canonizations. These, however, would not become a sole papal prerogative until the thirteenth century.

If there is a case to be made for when the pope emerged as the Pope, or at least what modern folks imagine when they picture the medieval papacy, it would be during the great reform movements of the eleventh and subsequent centuries. For the sake of brevity, the reforms championed among popes, monastic foundations and other centers would attempt to unify and regulate the far-flung communities of believers and establish papal authority in the farthest reaches of Christendom. It is under the reform popes that we see our first ‘crusades’ and stronger appeals to a universal Christian community. Excommunication and the suspension of the sacraments –long-standing papal prerogatives, if rarely used—were also utilized more frequently and papal insistence on investing bishops with the symbols of their office, as opposed to the temporal princes of Western Europe, would spark its own controversy. It was not an easy transition and the conflicts between popes and princes that would play out over the next few centuries would see emperors excommunicated, popes bloodied, anti-popes raised, and much angst and gnashing of teeth all around. Papal developments beyond the eleventh and twelfth centuries, including the crystallization of canon law, are a bit outside of my wheelhouse though. Seems like a good place to stop as any :)

Hopefully the above gives you some sense of the transformation of the papa of Rome as the representative of Latin Christendom among the five patriarchs into the medieval pope. I apologize that I am unable to cover trends in Eastern Christianities in greater detail. The fallout of the various Christological controversies and later the Arab conquests of former imperial territories obviously had a significant impact on the dynamic between the patriarchates. Haha something to be aware of, even if I have not covered it here.