Do stars symbolize fate in Shakespeare?
This is a very good question.
It is important to realize that Shakespeare is writing about a certain context. Also it is important to know that Shakespeare is writing within a certain context. Both of these contexts are different than our context.
When we consider this, we are in a better position to interpret Shakespeare. With this said, stars do often represent fate. This is because in Shakespeare's time things like astrology were seen to be real and powerful enough to chart a person's life. This has always been the case since the ancient world. We can get a sense of this in Julius Caesar. In this play, Cassius, one of the conspirator says:
"The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings."
The point is that stars, that is, fate, should not dictate the lives of people. People should take control. Cassius has to make this point, because the assumption is that fate rules people's life. In light of this, yes, stars often times can be a way of speaking about fate.