I don't know why you believe that the set of naturals is finite. If it's finite, then there is a largest natural number N. If n is natural, then n+1 is natural. So N+1 is natural too, which is impossible since N is the largest natural number. That should be enough to say that the set of naturals is infinite.
The set of naturals is {1, 2, 3, 4, ...}. The set of squares of naturals is {1, 4, 9, 16, ...}. You are talking about associating the set of naturals with the set of square roots of naturals, {1, sqrt(2), sqrt(3), ...}, and this also works too. Nowhere is it required that sqrt(n) be a natural number. All you need is an association n <-> sqrt(n).
I'm not trying to avoid the question. I just don't know what you're getting at.
I'm not sure what the question is about the length of the line. Do you want me to say that it's infinite? Of course it's infinite, since the size of the naturals is infinite. Adding the reals makes the line infinitely longer (this is not obvious).