A tenet of the Christian faith (and I speak from an Evangelical point of view) is that, in the words of Christ,
"'God is spirit, and those who worship Him must worship in spirit and truth'" (John 4:24 NASB Updated).
The very suggestion that God somehow possesses in his being DNA with which he could cause the Virgin Mary to conceive, violates this key tenet of Christian faith. God, who out of the fullness of his being created the human race out of one man and one woman, cannot have DNA. He is, rather, the creator of DNA. In other words, DNA was God's idea.
The primary reason for the incredible complexity of the human genome is the infinite creative genius and power of God, who spoke the universe into existence by the word (or Word) of his power (John 1:3). The One who created all things did so when there was literally no thing besides himself (except perhaps the angels, though that is an entirely different matter).
In Psalm 94, the psalmist asks a good question. I'll include the context for the question:
"Pay heed, you senseless among the people; And when will you understand, stupid ones? He who planted the ear, does He not hear? He who formed the eye, does He not see? He who chastens the nations, will He not rebuke, Even He who teaches man knowledge? The LORD knows the thoughts of man, That they are a mere breath" (vv.8-11).
The psalmist expressed poetically a profound truth regarding the personhood of God; namely, the God who created the ear can also hear; the God who created the eye can also see. Hearing and seeing originated in the God who hears all and sees all. The senses of hearing and seeing are but metaphors, which originate in the eternal attributes of God. Our human ears and eyes, in other words, tell us metaphorically something about who God is.
Just as language as a platform for human understanding is filled with metaphorical expressions, such as "I see" as an alternate way of saying "I understand," so also in similar fashion the created, material universe is a platform for insight into the being of God, who is the creator and sustainer of all things.
"For since the creation of the world His [i.e., God's] invisible attributes, His eternal power and divine nature, have been clearly seen, being understood through what has been made, so that they are without excuse" (Romans 1:20 NASB).
In conclusion, the point I have been making is simply this: There is a great (infinite?) divide between who God is by nature and what he has created out of the fullness of his creative power and genius. While God is pleased with "man to dwell, Jesus our Emanuel,"
as the Christmas carol puts it, he was and always will be transcendent vis-à-vis his creation.
By the same token, however, God through his Christ is also immanent, since Jesus revealed God in all his fullness in the vehicle, so to speak, of an eternal "flesh and bone" body, the body of the God-Man (see Colossians 1:15-20). Exactly how God invested both divinity and humanity in the person of Jesus is, I believe, quite mysterious. That Jesus was conceived of the Holy Spirit is the reason why Jesus could become the perfect, sinless, and spotless Lamb of God who takes away the sin of the world (John 1:29 and 36).