Space is expanding carrying everything along with it. This is in addition to their intrinsic movement within space.
Photons flying out at the speed of light through space are also boosted by the expansion of space itself. This results in an apparent separation faster then the speed of light thought no violation of relativity has taken place.
No. Gravity waves are too slow. They only travel at the speed of light.
Some years ago some scientists built a spaceship, SOTI as I recall, that could travel many thousands of times the speed of light. One guy traveled out to edge of the universe where he encountered what he called a wall of bathroom tissue. Funny thing is every time he tried to head into it he found himself heading back into the universe but at the exact opposite end of the universe from where he started. Or so the story goes.
Anyway, for some reason SOTI has been taken out of service. As far as I'm aware she hasn't flown since the series ended.