I. See M. Olsson On Faltings’ method of almost etale extensions, chapter 5. He discusses there a version of this fact over a dvr, but I think you can easily extract what you want, if you really don't want to look at SGA.
II. Yes. One possible precise statement is: a smooth scheme $X$ over a field $K$ of characteristic zero admits a basis for the Zariski topology consisting of affine open subsets $U\subseteq X$ which are $K(\pi, 1)$ for the etale topology. Moreover, if $K=\mathbb{C}$, then each $U(\mathbb{C})$ is a $K(\pi, 1)$ as a topological space, and the fundamental group $\pi_1(U(\mathbb{C}))$ is a good group (in the sense of Serre).