Are simply connected open sets in $\mathbb{R}^2$ homeomorphic to an open ball?
Yes, this is the Riemann mapping theorem. You get much more than a homeomorphism: you get a biholomorphic map.
Yes. In fact, more can be said... The Riemann Mapping Theorem states that the homeomorphism can be taken to be biholomorphic (as a complex map), if $U \neq \mathbb{C}$. See this link for a much more detailed treatment and proof.
Hope this helps!