The set is what has the dimension. And the set under discussion (the Cantor set) is the set remaining after repeating this process infinitely many times. That is to say, every time you repeat the process, the set gets smaller; so, if you take the intersection of all the finite stages, then you get what's left after repeating the process infinitely many times. That remaining set is the Cantor set being discussed.
> Would it be different if we removed 1/4 repeatedly?
That would result in what's known as a "fat Cantor set". If I'm not mistaken, it would have Hausdorff dimension 1, rather than than something intermediate like the usual Cantor set.
> Would it be different if the line was longer than 0 -> 1?
No, the length of the starting line segment is not material here.