With regard to #1, I would plug in z = x + iy, take the magnitudes, and see what you get. Geometrically, you're trying to find all z such that its distance from -6 is greater than its own magnitude.
With regard to #2, I could be wrong, but it seems to me that f(z) does satisfy the C-R equations. Remember that your x and y are no longer independent! All your computations (which look correct to me, by the way) were set up to ensure that f(z) does obey the C-R equations.