Were Greek-speaking Byzantines reading the same set of canonical classical texts that we would list?
i.e.:
Would they have rated the "greatness" of these texts in the approximate order that we do? Did they have more contemporary texts that they would have rated as highly as these? Did they think any of these canonical texts were rubbish in comparison with our own appraisals, or controversial for being pagan?
In both Islamdom and European Christendom a kind of "scholasticism" was necessary to accommodate pagan philosophical texts to monotheism, did this exist or was needed in Byzantium?
I can't entirely answer this question, but I can partially give you an answer. Firstly, yes: the canonical classical Greek texts are very similar to our own 'canonical' set. Partly because texts that survived as canonical for us, survived because they were canonical for them.
However, texts received different degrees of emphasis and appraisal. One should not, for instance, neglect the 'Canon of Ten' Attic Orators who were widely read and used as models for Attic prose and rhetoric in the Second Sophistic.
Secondly, Christian texts came to play a very strong role in both (a) 'canonical' literature of the Byzantine period, and (b) transmitting classical (pagan) philosophy. The clearest example is that of Gregory of Nazianzus, whose texts were second only to the Bible in being copied and transmitted in the Byzantine era. Writing such as his shows the Late Antique synthesis between theology and classical philosophy traditions. It was more often this kind of synthesis that survived into the Byzantine era, though Aristotle was probably more influential than Plato directly (i.e. people tended to read and use Aristotle directly, they tended to utilise Plato through the canons of Neo and Middle Platonism).