About the two circles. Here is a plan: just send the point $-2i$ to $\infty$ somehow. For instance, by $z \to 1/(z+2i)$. Then look at what happens to the circles. They will map to two parallel lines, but probably not the lines that you want. But this is not a problem: you can simply add a second transformation that will map the two lines that you have to the two lines that you want.
As for the second question, it probably does. Have you tried it?
UPDATE: Here is a more detailed answer to the second question. We have four distinct concyclic points and we need to prove that there is a Möbius transform that sends them to $\pm1,\,\pm k$ where $k \in (0,\,1)$. Here is a rough sketch of a proof.
Step 1. The points are concyclic, so there is a Möbius transformation that maps them all onto the real axis. So we can assume from the start that our 4 points are all on the real axis.
Step 2. We have 4 points $z_1,\,z_2,\,z_3,\,z_4$ on the real axis. Prove that there exists such a $k\in (0,1)$ that two cross-ratios are equal:
Technical detail: you might need to relabel points $z_i$ in a different order to do this.
Step 3. Now you can use the fact that Möbius transformations can be specified by three points. So there is a Möbius transformation $\omega$ such that $\omega(z_1)=k,\,\omega(z_2)=-k$ and $\omega(z_3)=1$. And you can prove that $\omega(z_4)$ will be automatically equal to $-1$ using the fact that $(z_1,z_2;z_3,z_4)=(k,-k;1,-1)$. This is it.