Yes it does make her the Mother of God.
Motherhood is both an act and a relationship.
Mary became the mother of a divine person when she conceived him in her womb, carried him for 9 months, and gave birth to him. She has the relationship of a mother to a divine person and will have that relationship eternally.
Nope, and this isn't what the Catholic church originally taught either.
We should distinguish the term
Theotokos from
mother of God, because there is a subtle yet important difference. The term
mother of God could be taken wrongly as implying that Mary was the source or originator of God, similar to how Juno was the mother of Vulcan in Roman mythology. Of course, Christianity teaches that God is eternal and that Jesus Christ has a pre-existent, divine nature. The idea that Mary is the mother of God in the sense that she was the source of God or somehow predated God or is herself part of the Godhead is patently unbiblical.
The term
Theotokos, on the other hand, is more specific and less open to being misconstrued.
Theotokos simply implies that Mary carried God in her womb and gave birth to Him. Mary was the human agent through whom the eternal Son of God took on a human body and a human nature and entered the world. The term
Theotokos was a succinct expression of the biblical teaching of the Incarnation, and that is how the Council of Ephesus used the word. Mary is the “God-bearer” in that within her body the divine person of God the Son took on human nature in addition to His pre-existing divine nature. Since Jesus is fully God and fully man, it is correct to say that Mary “bore” God.