“Mother of God implies existence before God.”
No, it doesn’t. It only implies Jesus is God. It can’t imply pre-existence before God because May is only a creature, and not the Creator. The Creator, however, chose to be born as a man (a man who was also God) and to be carried into the world by a woman. Women give birth to persons, not natures. Jesus was a Divine Person. Mary could only be the mother of God for that is what Jesus is.
No it doesn't because if someone had never heard of Jesus, they would not know that *mother of God* meant *mother of Jesus*.
There is simply no way for anyone to deduce that from the term unless they had been indoctrinated by Catholicism.