The argument that *Mary is the mother of Jesus and Jesus is God therefore Mary is the mother of God*, leads to the following conclusions using the same (for lack of a better term) *logic*:
If Mary is the mother of God and God is the Trinity, then Mary is the mother of the Trinity.
Your logic is incomplete. Mary is the mother of Jesus and Jesus is God in the Second Person of the Trinity. As the Second Person of the Trinity, Jesus is God, whole and entire. This is His divine nature. As the child conceived of and born from Mary, Jesus is a human being. This is His human nature. Jesus is true God and true man. As the mother of the person Jesus, Mary is His mother in His divinity and in His humanity. Therefore, Mary is the mother of God.
If Mary is the mother of God and God is the Trinity, it does not follow that Mary is the mother of all three persons of the Trinity. As explained above, and as revealed by the Holy Spirit, Mary became the mother of the Second Person of the Trinity. As the angel Gabriel foretold, “The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God.” The Holy Ghost and the Highest were not born of Mary, just the Son.
As you noted, the Holy Spirit called Mary the mother of Jesus. But He also called her “his mother” and made many references to how she conceived of and gave birth to a divine person and that this divine person was her Son. I will not repeat all these references here, but I will ask again:
Can you explain to me how Jesus can be separated from God with regard to His mother?
Mary is not the mother of His divinity.
His divinity pre-existed Mary, it did not come into existence when Mary conceived Jesus.