>doesn't make it any more or less likely that you're going to hit anything.

I believe that would be wrong. Also not an astrophysicist but I would guess that most of the stuff in our galaxy is moving at speeds which are, on average, relative to our galaxy, and moving at some average rotational speed around it. Of course there could be stray bits moving quite quickly on through.

So moving at c relative to the planets and solar systems swirling around our galaxy probably would make you more likely to hit something.