What elements of tragedy and comedy are there in The Tempest, and how does the play argue that justice should be tempered by mercy?

Lots of questions here. Though it does, just about, observe the Aristotelian unities of time and place (that is, the play takes place - nearly, at least - within one day and in one location), I don't think you can classify "The Tempest" as a tragedy. Nobody dies, and it ends in a marriage: which is, of course, usually seen as the definition of comedy.

But there are elements of tragedy in "The Tempest": the whole slavery question (Caliban, Ariel, and potentially even Miranda as slaves to Prospero - the argument that Prospero, usurped from Naples, usurps Caliban of his island, and so on), and - of course - the implication that Prospero will take his revenge on his brother for the way he has been treated.

That Prospero eventually forgives his brother - and tempers his desire for justice and vengeance with mercy and grace - is usually described as an entirely positive move, and, for the resolution of the play, it is. But have a look at Prospero's final epilogue: not the words of a man blessed and enlightened by his own forgiveness, but a worried, ambiguous speech that foresees an ending in despair.

Justice or forgiveness? Shakespeare leaves it very, very ambiguous.