Hi Tom; I meant statement 1 (the category of sets is an ordinary topos, rather than an $\infty$-topos). Statement 2 would be a special case of the blanket hypothesis "homotopy colimits in X commute with homotopy pullback". The more general property you describe also holds in any $\infty$-topos.

The homology equivalence of the identity component of QS^0 with the classifying space of the infinite symmetric group gives a homology equivalence of its universal cover with the classifying space of the infinite alternating group G. Hence $\pi_2 QS^0 = H_2(G)$ is the Schur multiplier of G: that is, the kernel of the universal central extension of G (which is also the universal central extension of A_n as soon as n is reasonably large; I think it starts at n=8.)

"Sheaves in Geometry and Logic" by Moerdijk and MacLane is a pretty good read (as is Uncle John, but I've never seen topos theory in there).