Well, I'm not wanting an argument but I'd like to try explaining something about modern physics.
To ask whether it's possible to travel faster-than-light is a mistake. The question implicitly assumes that speed can be defined as distance travelled / time elapsed, but that's a definition from Newtonian physics which isn't valid at high speeds. In modern physics the formula is more complicated and it would be better to ask "Do speeds faster than light exist?" Since space and time won't behave in the ways suggested, the answer is that there is no valid calculation involving a distance and a time interval which can give a result bigger than the speed of light.
Asking whether something is possible is loaded; in recent decades we've become used to amazing things becoming possible. Asking whether a valid calculation exists gives a better picture of the real issue in physics.
That's the best I can do without bringing maths in.
As for the Alcubierre drive, it may be that this sidesteps the issue and lets you arrive somewhere without apparently having to take a direct route, but it only just functions as a piece of game lore. When you decide to compress the space of half a solar system to traverse it quickly, what happens to all the inhabitants of planets in that half? And what happens to another ship travelling at right-angles to yours? It's a neat piece of handwavium, but there's no way to get a rigorous theory out of it which will have the effects we want.