Regarding the quotation, is tragedy a way of dealing with religious topics outside of religious faith? Or is religion an answer to a the tragic sense of existence?
Religion seems to hover in the background (and often the foreground) of tragic literature. The reason seems to be that both religion and tragedy are concerned with the ultimate place of man in the universe. Moreover, when we consider the Abrahamic religion (roughy, the Old Testament half of the Bible) we could say that the tragic sensibility stands at the very heart of the religious vision, indeed even inspires the religious text. The Book of Job is a good example. But so are the first chapters of Genesis.
This is a good question and it can be answered in many different ways. In other words, there will be differences of opinion. Let me offer one thesis. In a broken world, tragedy oftentimes leads to theological reflection. In other words, religion seeks to give an answer to tragedy.
When we look at the Greek world, we see this point very clearly. Greek tragedies were performed at religious festivals. Moreover, one of the principle themes of Greek tragedy is to avoid hubris. In other words, people need to realize that they are not divine and so they must know their place in life. Even the Oracle at Delphi had the maxims: "Know thyself" and "Nothing in excess." Basically, the point was know that you are not divine and so do not transgress.
Second, when we move over to the Hebrew Bible / Old Testament, we also see many tragedies, like the Book of Job, as you mention. These times of tragedy show the need for God. So, according to the Hebrew Bible / Old Testament, suffering and tragedies do not lead to unbelief in God, but greater belief in God. This is why the psalmist could say in Psalm 119:71:
"It was good for me to be afflicted
so that I might learn your decrees."
In light of these two point, we can say that religion is an answer to the sometimes tragic existence of life.