RESOUNDINGLY YES!
He was God before He came to earth.
Perhaps this is why God doesn't ever refer to Mary as "mother of God."
Everything you said here is true. Mary was not His mother before He came to earth. He existed from all eternity co-equal to the Father and the Holy Spirit, consubstantial, or as some put it, "one in Being" with the Father and the Spirit.
But she became His mother at a point in earthly time: when He was conceived in her womb. She is the mother of a Person. That is to say, she conceived, carried, and gave birth to someine. Who was that "someone?" Jesus. It does not mean she was the origin of his Divinity, that she's some kind of big ol' goddess, or that shes older than God or anything like that.
I think the reason people have his misconception, is that they don't know the history of the term. The Church adopted the term in order to combat the heresy that there were two Jesuses, one God and one Man, and that the Jesus who was born from her was not God. (But that's false. He's not two people.)
He existed through all eternity. She conceived--- gave flesh to him ---, gestated, and gave birth to Him only "at a point in time." Not from all eternity.
Obviously, Mary was, like us, a human person. Like us, she did not exist before her earthly life, at all.
She did bear God in her womb, however. Jesus is God.
There is no reason why any Christian should hesitate to call Mary the Mother of God.
"Who am I that the mother of my Lord should come to me?" said her kinswoman Elizabeth. And was not this Lord her God?
Now you'll drug around the pond for responding to THIS bait!