The definition of marriage as you put it specifically denies rights and privileges from a certain segment of society. That is un constitutional. Simple as that.
to show causation, but correlation yes. And there's a logical reason to believe there might be some degree of causation in *some* causes.
Hate crimes existed before (of course), and they would still exist with or without Prop 8. The question is does the quantity increase because of the vote, and it's possible the answer is yes.