Is Mary, "Mother of God," the mother of the Son or of the whole Trinity? I find the Catholic title "Mother of God" for the Virgin Mary confusing. It is clear that she was the mother of Jesus, the Son. But the title suggests (indeed, not literally, but still) that Mary is mother of God as a Trinity. Is that correct? If so, it creates the problem that a human (or is there reason to say that Mary wasn't (entirely) human?) gave birth to God, while God created mankind.

This problem doesn't exist when Mary is only mother of Jesus, because then it could be merely a way of speaking to say that Mary was the one through whom the Word became flesh, which would be the Protestant view as described in Why do Protestants not refer to Mary by the title &quot;Mother of God&quot;?
But the title seems to refer to God the Trinity. Is that correct? 

No it does not.

First trinity is not God. Trinity is how God is. That is trinity is not a person, it is a relationship. So Mary cannot be the mother of Trinity.

Secondly it would be nice if you know the context why and when this was declared as a dogma. The problem rose when Nestorius said the Mary held within herself only the human Jesus and not divine Christ. That is Mary is the mother of human Jesus. But as we all know mothers don't give birth to "human nature", but to a person. Hence Mary is the mother of a person named Jesus, who happens to be divine in an Hypostatic union. Since Mary is mother of His humanity, she is also the mother of His divinity.

When we say Mary is the mother of God, we do not mean that she is the author of His Divinity. But we mean that she held in herself and gave birth to God and Human Jesus.