The emperor controlled the church and the empire essentially formulated a doctrine where the emperor was a divine monarch, appointed by god. Since he and everyone else genuinely believed that the unity and orthodoxy of the church was a predicate for the continued security of the empire they meddled constantly. Since Constantine the emperors have been deeply involved in the affairs of the church, constantly trying to enforce orthodoxy and ensure that the church did not fall apart, as it was threatening to do almost constantly. First the Arians (is god the son inferior to god the father?), then the Nestorians (how united are the human and divine natures of Jesus and what about the Virgin Mary?), then monothelitism (does Jesus have one energy or two) then Iconoclasm (is praying to icons idolatry?) and so on and on and on. Trying to maintain orthodoxy occupied a huge amount of the emperor's time and energy after the empire converted to Christianity and these doctrinal controversies would often go on for generations until a compromise solution could be found. This does not even mention that the individual patriarchates would often have their own little squabbles and would constantly throw out excommunications and interdictions like flowers at a wedding.

An effect of the emperor constantly striving for doctrinal unity was a slow and inexorable drifting away from the Bishop of Rome and the western church as a whole. Rome and Constantinople were often in a state of schism and Rome began to develop its own theology and ideas, speeded up by differences in language and culture and missionary efforts among the Franks and others. By the time that the final break came in 1054 the two churches were really only united in name only. Further, the collapse of the temporal power of the empire ensured that the papacy was basically out of the power of the empire.

As far as the personnel occupying the patriarchate of Constantinople was concerned, the Emperor could basically hire and fire him as he wished. Justinian had one of his patriarchs (Eutychius) deposed and him and he was then put back in the position after Justinian's death. I can't remember exactly what powers of appointment the emperors in Constantinople had over the other patriarchates and I don't want to spout my mouth off, but they were quite extensive.

As for what happened after the end of the empire, most of the really important doctrinal stuff had been settled by this point -- there was simply less of a need for constant meddling on a pure doctrinal level. As for political involvement that was as deep as ever. But this is something that 100 books could be written on.

While it is a bit out of date now I would suggest you have a read of the Formation of Christendom by Judith Herrin. It is a good introduction and as a book on the history of the early church goes it is quite entertainingly written.

I'm assuming that by "Great Schism" you mean the formal 1054 split between the Latin and Greek churches, which had virtually nothing to do with iconoclasm and only a little to do with the role of the Byzantine emperor. While there had been some conflicts off and on for centuries, there had never been outright schism until 1054. The western church in fact had very little involvement in the iconoclastic controversy, which in any case had by 1054 been over for two centuries, and settled in favor of icons.

Tensions between the Greek and Latin churches had been escalating for some years before 1054 for a number of other reasons, like a struggle for control of churches in areas like Sicily that had both Greek and Latin believers. The decisive event, however, was Pope Leo IX's demand, communicated to the Patriarch of Constantinople by legate in 1053 and also by letter in 1054, that the Greek church recognize the supreme authority of the popes. Leo IX was a very energetic reformer, but his attempts to enlarge papal power were, in this case, ill-advised. His choice of legate, Humbert of Silva Candida, was also unfortunate because Humbert and the patriarch, Michael Cerularius, were equally stubborn and outspoken. Cerularius had been very outspoken about his opposition to the Latin church's form of the creed and use of unleavened bread for celebrating Mass.

While the relationship of the Greek church to the Byzantine emperors would no doubt have made Leo's demand even less palatable, it hardly seems likely that Cerularius would have acceded to it anyway. Leo IX did advance in his letter a particularly vigorous interpretation of papal supremacy, including the claim that Constantine had in fact ceded authority over the Roman Empire to the popes, the so-called "Donation of Constantine." In that sense, it's possible to regard the 1054 split as related to the authority of the Byzantine Emperor. However, it was much more a diplomatic failure. Humbert, Leo's legate, delivered a bull (official papal letter) excommunicating the Greek church right on the altar of Hagia Sophia after failing to persuade Cerularius to accept papal supremacy. This move was hardly likely to sway Cerularius, who declared the entire Western church anathema in response.

Greek objection to the Latin church's form of the creed (the "filioque controversy") and papal demands for supremacy over the Greek church remained the most significantly divisive issues long after 1054. If the emperor's relationship with the Greek church had been a significant point of contention, then the Greeks would surely have accepted the 1435 union of the Greek and Latin churches that the emperor himself brokered with the pope at the Council of Ferarra-Florence. Instead, the great majority of the Greek people and clergy rejected the union on the same sorts of doctrinal grounds that had precipitated 1054.

I know this doesn't directly address your question about the role of the emperors in the Greek church, but I hope it's helpful nonetheless since it does address the premises on which your question is based.

Sources: E. A. Livingstone (ed.), Oxford Concise Dictionary of the Christian Church, 2006 William R. Cook & Ronald B. Herzman, The Medieval World View: An Introduction, 3rd, 2012