Though technically a warp drive which stretches/compresses the fabric of space isn't a FTL drive in the absolute definition because you aren't physically trying to move faster than light.
People love to bring that up, but that the ship itself is not physically moving faster than light relative to the bubble doesn't mean that the bubble and ship aren't moving at FTL speeds as seen by an outside observer (granted, you don't see the bubble, but still). The bubble is just there to facilitate movement at FTL speeds without dealing with acceleration to/past lightspeed. It's a separate problem.
And yes, seeing an unexploded star that you watched explode breaks causality. In the simplest case, let's say you warp from the exploding star to another faraway star around which your friend orbits. From your perspective, the star exploded, you left (after it exploded, having observed the light from its explosion), and you arrived at their star. From their perspective, given a sufficiently powerful telescope, the sequence of events is reversed: you arrived, you left, and then the star exploded. Causes cannot follow effects. Now, you can see unrelated events happening differently from different perspectives just fine; to return to our exploding star example, let's say we have four stars in a line A-B-C-D and stars B and C explode simultaneously. A would see B explode followed by C; D would see the reverse. That's allowed by physics.
However, if A launches a (subluminal) ship to destroy stars B and C, all the stars will see that ship launch (and arrive) before stars B and C explode. The same cannot be said of a superluminal ship, and that's what breaks causality.
This isn't time travel as portrayed in science fiction, no, because science fiction writers would rather write good stories than physics textbooks.
Regarding light cones, by the way: it's a ball (that is, a solid sphere) at any one time, but it looks like a cone in Minkowski spacetime and that's what matters here. If you were to look at 2d space and display time along the third axis, as you watched the ball expand and move along time its edges would describe the light cone.
EDIT: If you just wink the light on and off, then yes, it's a sphere at any one time, but for our purposes it might help to think of the ball of space that knows you turned the light on rather than the sphere currently experiencing the flash.
