Well it depends. The Byzantine emperor was the unquestioned head of state and according to official imperial ideology he was appointed by god himself. But by that definition many other monarchies like the kingdom of France or the Holy Roman Empire were also theocracies. That wouldn't be wrong per se, as there was definitely a theocratic component to the rule of their monarchs. The kings of France and England for example were said to be able to heal certain illnesses by their holy touch and the Byzantine emperor automatically held a clerical rank equivalent to that of a deacon.

But in the east as in the west there were much more prominent holders of religious authority in the form of the christian clerus. In the Byzantine sphere of influence their head was the Ecumenical Patriarch of Constantinople. Normaly he was appointed by the emperor but this didn't necessarily make him a puppet of the ruler. There are many cases of patriarchs or other parts of the clerus defying the will of the emperor especially in matters of religious law.

On the other hand the clerus wasn't really involved in governing the Byzantine Empire. Much less so actually than the bishops of the west that often could hold wordly fiefdoms. The empire made use of a state bureaucracy that was quite sophisticated for its time. Constantinople was the seat of several civil servants responsible for different resorts like the treasury or the support of the army. The provinces were governed by officials appointed by the emperor. Of course over the course of over a thousand years of existence the details of how that system exactly worked, how those officials were called and what their responsibilities were changed quite a bit.

So there clearly was a dividing line between worldly and religious authorities in the Byzantine Empire. That line gets a liddle blurred when it comes to the person of the emperor himself but not enough to really call it a theocracy in the same vein as let's say the Vatican or the ancient israelite temple state.

In no way. The Byzantine Emperor was the head of the empire , but he had the right to appoint the Patriarchs (heads of the churches). The Patriarchs had little to no political power. Contrasted with the Holy Roman Empire where it was the Pope who crowned the Emperor, but that also wasn't a theocracy because you still had an emperor as head of state.

In order to be a theocracy, you have to have an actual god or that god's priests as effective heads of state. Neither the Byzantine nor the Holy Roman Empire had those.