Regarding Q3, one more explanation of why $\pi_2(G)=0$ (which is similar to what Claudio Gorodski points out above) can be found [here][1]. A nice observation made there is that one can use the same arguments to show that **$\pi_3(G)$ is torsion-free!** (which I guess was not in your list of facts about homotopy of compact Lie groups).


  [1]: https://mathoverflow.net/questions/8957/homotopy-groups-of-lie-groups/8961#8961