To take your second question first:
was Mary Jesus' biological mother, or was she a surrogate?
According to the vast majority of Christian churches, Mary was in every way the mother of the human being Jesus Christ. So yes, she was his biological mother. He was genuinely Mary's son, as you see confessed time and time again in the creeds, confessions and liturgies of pretty much every church.
Your second question:
Was Jesus made from God's Spirit "fertilizing" Mary's egg or was he a being entirely created through God's spirit but merely placed in Mary's womb?
The best answer to this is "neither".
According to Nicene Christianity, the Son of God, the second Person of the Trinity, existing from eternity, of one substance with the Father and the Holy Spirit, took flesh in the person of Jesus Christ and was born of the Virgin Mary. (See the Niceno-Constantinopolitan Creed.)
So the Son of God existed before the conception of Jesus Christ. When he took flesh, he received his human nature from the Virgin Mary. (See, for instance, this agreement between the Catholics and an alliance of evangelicals.)
In my personal opinion, thinking about this in terms of eggs, embryos and DNA is not a helpful way of addressing this mystery, but your mileage may vary!