I'm assumed that it's called C space because the speed of light is normally termed c - so the space was described in terms of the limiting speed.

wouldn't that break causality ? because light / information would still travel in 'C space'.

Not really. Sound travels faster than light so we see really fast moving objects before we hear them. If we folded the universe and took a shortcut then we'd just arrive at the destination before it's visible that we've left. But that's little different to seeing stars millions of light years away shining away which would have already died thousands of years ago (hence why some scientists romanticize about astronomy as being like time travel - as they're often looking at events that happened hundreds or even millions of years in the past)

In any case, there's already enough literature on this subject to last you a life time so if you have any question then it's an interesting subject to read. And by "literature" I don't mean "Star Trek" (though their "Warp Drives" are based on a similar premise or warping 3 dimensional space - despite obviously being a work of fiction)

Yes. Of course the only evidence we have for causality is that we have not observed it being broken (and recognized that observation)