We do not say that God dies on the cross, but that Jesus who is truly man as well as truly God, dies on the cross. Jesus does not die in His godliness, but in His humanity.
Mary is Mother of Jesus who is God as well as man. She is the mother of His humanity.
We say Mary is the Mother of God. Jesus is one person with two natures. Mary is His mother, The Church says Mary is mother of God. That is a title. While He took His human nature from her, we do not say she is the mother of His humanity. That is not her title. She is the mother of a person, the God-man. She gave birth to this person.
A mother is not the mother of a nature. She is mother of a person.
The Incarnation is the greatest mystery of faith. A mystery is something our finite intellects can not fully understand. There are many things like this. What is love? Why do we feel it sometimes and not others? We say the soul is the life of the body. We can be present when someone dies. They are alive one minute and gone the next. The soul leaves the body and they are gone. How it happens is a mystery.
How God could be God the Supreme Being, Infinite, without beginning or end, and enter His creation and become man as a few cells in a woman’s womb is a mystery. He who can not be contained is contained in a woman, a living tabernacle.
Why God would save us is not a mystery. He loves us and pities us. The mystery is the way He chose to save us. He is all powerful. His word spoke all that exists into existance. He only has to say the word and His will is done. Why would God become one of us, take on our nature and allow His creatures to torture Him to death?
He is showing us something about Himself that He wants us to know. We say He died for our sins. It is personal. He died for your sins and mine. That means I am the one who tortured Him to death. My sins killed Him and left Him dead in His mothers arms. The Jews and Roman soldiers were my agents.
They drove the nails, but my sins caused it.
We think of God as all powerful and invincible. I am a puny nothing. How could my sins, the evil I do, hurt this being of infinite majesty and power?
He shows us by becoming Incarnate and dying on the cross. The message is, see what your sins do to me. This is how much it hurts me to see my beloved creatures do evil, to murder, to rape, to lie and steal from one another, to hate.
His will is that we repent of the evil we do and be saved by His mercy. His will is that we repent of the evil we do and learn to love as He made us to love.
We can not see God. He became one of us so we could see Him and hear Him, and we murdered Him in our evil. He did this in hopes that when we realize what our sins do to Him we would repent and turn from evil.
After the Annunciation/Incarnation, the second greatest mystery is the Ascension. He did not leave His human nature behind. He is fully God and fully man eternally. He took His body with Him to heaven. How can that be? He still has feet and hands of flesh and they bear the marks of His Passion. He still has a heart and it beats. And He feeds us with His flesh, the bread of life.
We believe all these things and they are mysteries.