Part 1

Oof, oh boy! That is a bigggg question. Judging from your follow up, I take it to mean that you are interested in the role of ecclesiastic and religious institutions in the dissemination of Christianity in the Latin West during Late Antiquity and the Early Medieval Period? Well the answer is: they did quite a lot! It is hard to fit five centuries of ecclesiastic activity into one /r/AskHistorians post, but I will try to hit on some major activities and some of their developments over time. While ecclesiastical communities became increasingly organized between AD 300 and AD 800, it is important to remember, as others have noted, that the Church was not a monolith during Late Antiquity and the Early Medieval Period, and diversity of belief, practice and policy is the rule rather than exception for the Christianities of your question.

Right off the bat I am going to have to apologize. What you’re asking covers a lot of ground and demands more than its fair share of context to explain properly—more than can be conceivably fit into one answer. Any post that addresses five hundred years of history is going to miss some stuff and lean on common trends rather than specifics, but hopefully I can give you a glimpse of ecclesiastical activity over the course of Late Antiquity and the Early Medieval Period—though I know less when it comes to the Roman Church of fourth and early fifth centuries. However, if you want a great place to start reading, I second /u/cerapus's recommendation of Peter Brown's The Rise of Western Christendom. If that is a little too dense, the relevant volumes of Cambridge History Series on the History of Christianity are an excellent introduction and overview for undergraduates and non-experts. If you are a university student, it is likely that you have access to the online version. That being said, let's jump in!

The structure of the late ancient and early medieval church is probably as good a place to start as any. By the late third century the basic organizational hierarchy was in place, but over the course of the fourth, the infrastructure and economic capital of the church grew drastically as it assumed a prestige position within the social order of the Roman Empire. In theory the church was organized into bishop-led dioceses, whose boundaries usually corresponded to those of local civic administrations. Metropolitan bishops in turn oversaw the bishops of an ecclesiastical province --or in some cases, provinces—but even in the fourth century, this position was not universally used. By AD 400, four patriarchs --Rome, Constantinople, Antioch, Alexandria-- were widely recognized as having primacy over their own spheres of influence, with Jerusalem added as fifth in the AD 451. The Roman see, located in the old capital and surrounded by the tombs of countless saints, was already afforded a special position within the Empire, but at this point, the papa of Rome exercised little, practical authority over the West.

Again, while the church was organized it was still decentralized, especially in the Western Mediterranean, a region that largely lacked urban infrastructure—well at least compared to the East. While we cannot speak of a ‘Western Church’ at this point, we can still see the beginnings of a shared religious and cultural tradition that distinguishes it from the East. Part of this falls along pre-existent social and cultural divides. Latin became the liturgical language in the West as opposed to Greek. Even during the fourth century, the Western bishops would be underrepresented at the great ecumenical councils in the East. In subsequent centuries, the West would not experience the same discord born of theological disputes that rocked the East. Especially as ecclesiastics in the West confronted the issues of a disintegrating imperial administration, the structure and focus of western churches would take a much different direction from their colleagues in the East (which was itself undergoing change).

The collapse of imperial authority in the West disrupted the infrastructure of the church, but it survived, and indeed became the vehicle through which many late ancient traditions and institutions would be preserved. The formation of post-Roman successor states and the eventual conversion of their kings to orthodox Christianity had a corresponding affect on the orientation of ecclesiastical affairs. The metropolitan system of late antiquity largely disintegrated, and from the sixth century onward bishops were increasingly caught up in the orbit of their respective kingdoms. The appointment of new bishops was largely the prerogative of the king and regional councils (synods) were largely caught up in the affairs of the realm, enforcing ecclesiastical purity and educating the priesthood. Likewise, the role of the bishop in the administration of civic duties increased over the course of the fifth and sixth centuries, even as urban culture underwent its own changes, reorienting along new avenues of power.

I am skipping over quite a bit here, and am painting in large strokes, and its important to remember that the experience of bishops varied from region to region. I am largely ignorant of the organizational developments in North Africa, and of course, the contraction of ecclesiastical infrastructure in Britain contrasts with ‘Catholic’ Gaul or ‘Arian’ Italy. I am happy to elaborate on specific regions, but for the sake of brevity, let’s be content for now with the picture of a decentralized, regional church in the West, connected culturally, but largely concerned with local affairs.

Can you clarify what you mean by "the Church" as an entity distinct from Christianity?