Why did God choose Mary to be the mother of Jesus? Why was she so young?

​These two questions are actually hard to answer with accuracy. In many ways, the answers remain a mystery.  
 
From a theological perspective, we can say that God chose Mary to be the mother of Jesus because she, herself, was the Immaculate Conception. This means that she was the only fitting mother for God in the flesh. Mary was conceived in the womb of her mother in a miraculous way in that she was conceived without sin. God chose to bestow upon her a “preservative grace,” meaning, God preserved her from all stain of sin, including Original Sin, at the moment of her creation in the womb of her mother. Of course, He did this so that she would be a fitting vessel for God the Son to become incarnate in her womb. The grace that preserved her came from the Cross of her Son Jesus, but it transcended time so as to free her at the moment of her conception. Thus, her Son was her Savior, even though He had not yet been born in time. If that’s confusing, try to ponder it for a while. It’s a great mystery of faith and a deep one also.  
 
Additionally, Mary chose to remain free from sin throughout her life. Just as Adam and Eve were born without sin, so was Mary. But unlike Adam and Eve, Mary never freely chose to sin throughout her life. This made her the perfect vessel for the Son of God. Her body and soul were perfect, making her the perfect instrument.  
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But this only answers your question from one perspective. You might also be wondering, “But why Mary?” This is a question that is hard, if not impossible, to answer. It’s most likely a matter of God’s mysterious will. Perhaps God, who can see all things and knows all people even before they are born, looked at all women from all time and saw that Mary was the one who would never freely choose to sin. And perhaps for that reason God chose to gift her with the Immaculate Conception. But this is ultimately a mystery of faith that will be revealed only in Heaven.  
 
As for your second question, “Why was she so young?” this might be easier to answer from a historical perspective. Today, in the twenty-first century, it’s uncommon for a fifteen-year-old girl to marry and have a child. But it wasn’t that way back then. When Mary had Jesus, she was seen, not as a dependent child, but as a young woman ready to start a family. So it’s always important to try to understand the culture of the time when considering questions of history.