Relationship Between the Emperor and the Imperial Church

I think you should pose this question to one of the orthodox Christian subreddits, where you are more likely to get a complete answer on this topic.

All I will say for this is that, at least from Constantinople, is the idea of the ‘synergeia’ between the church and the state, the double-headed eagle of ‘Byzantium’ in which the Emperor is the protector of the church and the Christian peoples.

An excellent book with in information on your topic placed in a broader context is “Through the Eye of a Needle: Wealth, the Fall of Rome, and the Making of Christianity in the West, 350-550 AD” by Peter Brown (Princeton 2013).

Nicaea was the first ecumenical council. Effectively, Constantine created 'the Imperial Church' - he was the one who originally empowered the bishopric by giving them the equivalent powers of judges.

The Western Empire didn't even last a century, and was crippled before the first Western Emperor, Honorius, had even become an adult. It didn't have a real army from Day One. Within the first 15-20 years of his reign, Iberia had been mostly lost to the Alans, Suebi, and Vandals, Britain was evacuated of Roman troops, Burgundians and Franks were occupying Northern Gaul, and the Visigoths were settling Southeast Gaul after having sacked Rome. Within 40 years, Africa was gone, and Africa provided most of Western Rome's budget even before the losses of 390s, 400s, and 410s. Within 50 years, "Western Rome" was just Italy, Southeast Gaul, and modern Catalonia, with Roman isolated pockets in central Gaul.

The Western Roman Empire, in effect, didn't even have much temporal authority, let alone religious one.

In the East, the Emperor in Constantinople tried to enforce his rule over the Church starting with the Council of Chalcedon where he tried to raise the Patriarchate of Constantinople to supremacy, but with very poor results. The Alexandrian Church rarely listened, and broke off into the Coptic Church almost immediately after Chalcedon, taking along with it a quarter of Empire's Christians. The Antiochian Churches and Jerusalem Churches ran their affairs their own. The Emperor's Caesaropapist policy, solidified by the reign of Justinian, was only really enforced in the dioceses subject to Emperor-picked Patriarch of Constantinople and during the 2-century long Byzantine control over the Papacy in Rome(mid-6th century to mid-7th century), but even then, on theological issues, the Roman Church often defied the Emperor and even the Patriarch of Constantinople was known to defy him as well. The Imperial Court didn't really take control of the Church until the Lombards conquered Italy and the Arabs conquered Egypt and Syria, i.e Caesaropapism wasn't truly established until the Empire was reduced to Greece and Asia Minor.

Look for "history of papacy" books.

The Emperor viewed himself as the protector of Christendom (and really he was once the Muslims came on the scene).

The Patriarch of Constantinople and the Pope of Rome were both ostensibly bishops of the same church. However, the loss of the western empire left the pope separated from the eastern emperor and the church in the east.

Anywho, the eastern emperor was very pissed when the Pope named Charlemagne as Holy Roman Emperor and the Catholic and Eastern Orthodox Church officially separated in the in 1000s (known as the great schism). Pope Francis meeting with the patriarch of Constantinople a few years ago was actually the first meeting of the two in over 1000 years.

Some reading I’d recommend:

The New Roman Empire: A History of Byzantium by Anthony Kaldellis

Athanasius and Constantius: Theology and Politics in the Constantinian Empire by Timothy Barnes

Roman Imperial Policy from Julian to Theodosius by R. Malcolm Errington

Some of the ecclesiastical histories that cover this period talk about the relationship between emperors and church officials. From what I remember Constantine and his sons did not have that great of an understanding of Christian theology. They tended to follow the leads of their favorite clergymen, and some accounts basically argued that Constantius II constantly flip-flopped between theological interpretations based on who the last bishop was that had bent his ear. Eusebius is probably the best known early ecclesiastic historian, but he stops at Constantine, while Socrates Scholasticus, Sozomen, and Theodoret all covered the period after. 

If you have the time check out some of the history lectures archived here. There’s one specifically about Constantine, the history and evolution of the papacy, and many early Christian sects and schisms. They are presented by a Christian minister in a straightforward way. However he is a trained historian (medievalist) and does not approach his topics from an apologetic perspective.