To answer your first question, the inclusion
$A(G)\to S(G)$ is a homotopy equivalence. This is an application of Quillen's "Theorem A" (aka his "fiber lemma"). See Prop. 2.1 in his paper. To apply the fiber lemma, you just need to show that the fibers, which are those points mapping below a particular P-subgroup in S, are contractible. This means you need to show that the elementary abelian p-subgroups of a P-group form a contractible poset. Well, that's done by a conical contraction: just multiply each subgroup by the maximal elementary abelian subgroup of the center (to slide it up above the center)this characteristic subgroup), and then slide it down to the centerthis subgroup.
The Quillen Conjecture is definitely still open. Grodal showed some connections with higher derived limits (he shows that certain strengthenings of Quillen's conjecture are equivalent to statements about higher limits), and there's an old approach using finite topological spaces due to Richard Stong. There's been some interesting developments lately due to Shareshian (Hypergraph matching complexes and Quillen complexes of symmetric groups. J. Combin. Theory Ser. A 106 (2004), no. 2, 299--314) and others, but mostly they're about figuring out more specific information in particular cases, rather than attacks on the full conjecture.
In the early 90's, Aschbacher and Smith made a lot of progress (see their 1993 Annals paper). They proved the conjecture for groups not containing certain matrix groups as subnormal subgroups.
I don't know that anyone really has ideas for how to prove it in general.