How would Gregory respond to the following set of statements and questions? If Jesus is the son of God, that must mean he did not exist at some point. This does not make sense! How can the Nicene creed say that he was “begotten, not made”?! Furthermore, how can he share the same substance (essence) with God the Father? (You can bring in Athanasius in this if you like.)

The issue of Jesus not existing at some point was central to the Arian controversy and was settled at the Council of Nicaea in 325. Arians claimed "there was a once when he [Jesus] wasn't," and Eusebius of Caesarea and other anti-Arian bishops claimed that Jesus was co-eternal (of one substance) with the God the Father and existed in divine form before he was incarnate. The description "son of God" refers to his incarnation, not his divine nature, although due to the nature of divine time (kairos), his nature as the son is eternal rather than temporary. Finally, time, as we conceive it, is a result of our human limits. God (all three members of the Trinity) exists in eternity, seeing all time simultaneously spread out at once as we might see an entire computer screen at once.
For Jesus being begotten, not made, the creed is emphasizing that he was conceived and born to the Virgin Mary just as any other baby, albeit having been begotten by the Holy Ghost rather than a human father. This refutes the Monophysite and Docetist arguments that Jesus was not fully human and shows him both as fully divine and fully human.
For having the same "substance" as the Father, the main issue is that the Latin "substantia" is a translation of the Greek "ousia" and means "essence" rather than material "stuff' (which would be "hule" in Greek and "materia" in Latin). Thus, the claim is about the nature or inner essence of the Trinity rather than about their physical manifestations.