3. In what ways was the Byzantine Empire influenced by Christian and ancient Greek teachings and culture?

You're kidding, right? This is something you can write a book about...

Byzantine culture was established on Greco-Roman culture and Orthodox Christian teachings(which latter were often ignored in the social and political struggles).The Code of Justinian reflected the attempt to put together Roman Law and a Christian ethos. Byzantium communicated Greek culture,learning and literature to both Islam and the West(especially to Italy during the Early Renaissance)

the great emperor justinian of the byzantine empire was a christian and he based his laws (the code of justinian) on the bible and greek philosophy

all over the world you see a lot of philosophising long haired unemployed christians in the street... preaching burgshwa doo doo theories... and commandments.