Islamic terrorism. Why does it exist? Is there something inherent within the religion itself, or a reaction to Western activity?

It's a bit of both. Western powers have been meddling in the Middle East for centuries now, so there's a long history of events building up to the present that contribute. But the doctrine of the religion itself does also contribute. Nearly every major religion includes some grounds to justify war, even wars of aggression, but Islam is a more special case than most. The Quran was written by one guy in his own lifetime as he led his movement through war, into statehood, and into empire. The variety of situations he encountered often called for different measures, ranging from pacifism to aggression. So to make his actions legitimate, he wrote them into the Quran. It provides a singular timeline of justification for just about anything. Holy texts that were compiled over the course of centuries are a bit less susceptible to this effect. They provide a "that was then and this is now" kind of context. But the Quran is so compact that it hems everything into a single perspective, and hence detracts from others.