Is God the author of sin? If God is good, why does he allow evil?
The "author of sin" should be understood as a reference to primary causation. It should not be used in the sense of secondary causation - in other words, the agent or action itself. It is contrary to the ordinary course of what it natural to define authorship as only secondary causation when there is a prior metaphysical cause or a primary cause. God is the metaphysical author of sin (primary/ultimate cause), but is himself not sinful (not the secondary cause or the agent).
Thus, when discussing this topic, it should be considered under the subject of metaphysical causation and therefore should say that God is the ultimate/primary cause, while man (the one who does the action) is the secondary cause of sin. God can hate a thing in of itself for being evil (evil meaning disobedient, or contrary to himself) but may still yet while considering the consequences of allowing or willing such a thing to exist or come to pass for the sake of the greater good, still be called good - God's disposing or willingness to arrange and allow evil things to come to pass doesn't invalidate his goodness. In other words, God can hate evil and allow it to occur for the sake of the greater good while being the primary cause and still yet maintain his goodness and reserving the right to punish evil as evil.
God allows evil so that we may see the majesty and glory of his love and goodness. Evil is interlinked and a vital component for "freewill." Without evil, there would be no freewill, as there would be no other choice but goodness to be obeyed or acted out. And without freewill, there would be no possibility of love, because choice is intrinsic to love - it is not possible when it is coerced or forced.
God receives the most glory from a fallen yet redeemed world, opposed to the glory he would receive from an unfallen, "perfect" world.