Oh boy.

I am mostly surprised by this because my Internet-age upbringing has taught me that Christianity was always the enemy of intellectual development.

This is a popular sentiment on reddit, but one which is the product of reading back post-Reformation conflicts with the Catholic church (which is not the same as Christianity), and, later, many protestant sects, onto premodern history. In other words, the opposition between religion and science is a modern phenomenon. When the western Empire fell, it was the Catholic church that preserved Roman learning in the west; the modern university traces its lineage directly from medieval theological study; it was medieval theologians who preserved, rediscovered, and used the Classics.

Let's go through the quote you gave, seratim:

It was Christianity, not rational philosophy, that dispelled the swarm of lesser gods and godesses; the blood sacrifices and self-immolation; or the frantic resort to magic, fortune-telling, and divination

This is quite simply wrong. Well, I suppose Christianity did get rid of blood sacrifices, but only sort of since the eurcharist is a metaphorical blood sacrifice. Saints occupied much of the space once taken up by that "swarm", and magic, fortune-telling, and divination all remained quite popular, though their particular manifestations changed. We have stories of peasants burying the communion host in their fields to bring a good harvest, etc.

The emperor was held to be veritably a god. ... The Christian doctrine on this point went back to the saying gathered from Jesus...

And yet in the eastern Empire, the Emperor continued to govern the Church, and exercised control over the papacy into the eighth century. So even if this were true, it was true based on political circumstances, not some great enlightenment contained within all Christian teaching.

the Western world escaped from what is called Caesaropapism, a political system in which one person holds the powers of ruler and of pontiff. Instead, the spiritual power and the political power were held to be separate and independent.

This would have been an incredibly surprising bit of news to the later medieval papacy, which claimed absolute authority, both lay and religious, for itself.

But speaking in general of European history neither side has ever won out, and in the sharp distinction between the spiritual and the temporal has lain the germ of many liberties in later societies.

Also patently false. As the great medievalist Gerd Tellenbach once put it, "distinctions between religion, law, and morality are fundamentally anachronistic when applied to medieval Europe." At various points in time, both sides can be considered to have "won out".

At the same time the idea that no ruler, no government, and no institution is too mighty to side above moral criticism eventually opened the way to a dynamic and progressive way of living in the west.

And here our author reveals his biases: look at us, we're so great, modernity Yay! Gross.

I would note that this work seems to be part of a new wave of Catholic revisionist history, pioneered by Po-Chia Hsia (see his World of Catholic Renewal). This narrative seeks to counter the traditional story which places the Protestant Reformation as the motive force behind our glorious modern world by arguing that really, it was the Catholic church (here, "Christianity") all along. Of course, our modern world isn't all that glorious, and it was neither and both.

Some reading, if you're interested in a more nuanced view:

• Kevin Madigan, Medieval Christianity: A New History. I think the starting point here is to get a grasp on the historical development of Christianity, and this is a very accessible book.

• Jeffrey Richards, The Popes and the Papacy in the Early Middle Ages, 476-752. Much more of a specialists work, Richards provides an excellent insight into the dynamics of pope and emperor which served as the basis for later negotiations between lay and religious authority.

• Katherine Davis, Periodization and Sovereignty. This is a really dense book, but it's a great exploration of how the sorts of claims of discontinuity or continuity, such as those which we have encountered here, form the justifying mythology for our modern world. See also Timothy Mitchell's Rule of Experts for the application of much of what Davis discusses (and other medievalisms) in the British colonial context.

• Gerd Tellenbach, Church, State, and Christian Society at the Time of the Investiture Contest (p. 24 cited above). An old book, but a very subtle discussion of the relationship between Church and State in the era during the height of papal power.

I am taking Early Christianity to be Life of Jesus to about 500. In that case, it really depends on who you are and which way you are looking. There wasn't anything like a "consensus" as we would understand it today, because the Catholic Church didn't exist yet.

Constantine's legalization (or decriminalization) of Christianity of 313 came with some consequences for the interactions between church (as it were) and state. He inserted himself into church affairs by calling and participating in ecumenical councils. Some Christians, especially in North Africa, did not like the involvement of the Roman government (you can look into the Donatist controversies for further information, of which Augustine was a part). Theodosius I and Justinian I legislated their particular version of Christianity as the ONLY version, and were heavily involved in theological controversies.

To your comments: many clergy members were highly educated Roman citizens, and would have been familiar with philosophy (depending on where they were, it could have been Cicero or the Second Sophistic). If you're interested in the intersections between Greek philosophy and Christian elites, check out Sons of Hellenism, Fathers of the Church by Susanna Elm, or anything by the Cappadocian Fathers.

ETA: I have no idea why I am being downvoted - What can I do to be more helpful? FWIW Late Antique Roman Empire is my academic specialty, so I am kind of confused.