So I am the attacker and I have Hit and Run out as an objective. My opponent has one defender, I have one attacker. I play a fate card during the edge battle to kill his only character. My opponent put down Detained which has three icons and my fate card only has two icons. Do I win the edge battle since there are no more defenders, or does he win the edge battle since he had more icons?

Your opponent wins the edge battle. The only time an edge battle is an auto-win is if there are no defenders declared (and the attacker can still place cards in the edge battle if desired). In this case, the fate card killed the only defender, which means your engagement will ultimately be unopposed - but since the defender won the edge battle, you will not be able to resolve the edge-enabled icons on your attacker when it focuses to strike, and Hit and Run will not be able to deal its damage to the engaged objective.

Though you do not win the edge, you would get an unopposed bonus, assuming your opponent does not do something to remove your unit before the end of the engagment.