This is a theorem and proof from Conway's *Functions of One Complex Variable I*.

My main question is the questions that he follows the proof with and my understanding of the proof. It seems immediate to me why the theorem fails when G = C-{0} or when u(z) = ln|z|, but I'm not sure about my understanding of the first question. It seems to me that we need to ensure that any such region G needs to be big enough and "round" enough (i.e., having no strange corners or wavy parts) so that u can't move outside of G and the best way to guarantee this is to let G be either a disk or to be the entire plane. Is my reasoning correct?
Since the body text shows up a little weird on mobile I'll post my main question and attempt here.

Conway asks why this result won't hold when G is any general region. My reasoning is that we need to guarantee that G is large enough and/or "round" enough (i.e. the G has no strange corners or wavy parts) so that u won't escape G ~~(i.e., u(z) always falls in a ball contained in the image of G)~~ (i.e., that for every ball B(z,R) in G, there exists a ball B(u(z),ρ) in u(G)). Is my reasoning correct?
By the Riemann mapping theorem any simply connected region in C is biholomorphically equivalent to either the unit disc or C itself. So I don't see why a region being "wavy" would cause an issue for this theorem

My guess is that it's a topological issue of being open, connected, and simply connected. As you point out the theorem fails for the punctured plane

That being said I'm not sure why we are restricting to an open disc rather than any open, connected, simply connected region
I haven't read the whole proof, but something stands out very early on that the first integral in the proof assumes that you can integrate along a straight line in the region, which assumes the region doesn't have holes (which would be implied by convexity, say).