Was it possible that Christ could have been expected to speak Greek? - not in the divine gift of tongues sense, but as an ordinary person from Galilee?
Expected? Definitely not. While it is certainly possible that he knew a little koine, an individual of his social station and level of education would not have been terribly familiar with classical Attic Greek.
Aramaic, of course, was his native language.
Possibly. There were large communities of Greeks living in the Holy Land at the time, so he could have picked it up from them.
Jesus was a Aramaic Jew whose target audience was other Jews, not the local pagan Greek population. So while Jesus might have known some Greek, much like I know a few words of Spanish, he almost certainly didn’t speak it while preaching. Paul, on the other hand, being a well educated citizen of the eastern Roman Empire spoke and wrote in Greek.
The New Testament was written in Greek on the other hand because Greek was the common language of the eastern half of the Roman Empire, much like Latin in the west. Ironically, Christianity won a lot more converts from the pagan side of the fence rather than the Jewish side of the fence so it lost its Aramaic orgins pretty quickly.