simplicial stuff - Is there any generalization of the Dold-Kan correspondence?

Associated to a simplicial group $G_n$, there is indeed a "chain complex". Namely, for each n one has the group $C_n$ of all elements in $G_n$ that map to zero under the boundary maps $d_1, ldots, d_n$ (but not necessarily $d_0$) that are part of the simplicial structure. The map $d_0$ is then a group homomorphism $C_n to C_{n-1}$ with $Im(d_0) subset ker(d_0)$, and the image is a normal subgroup. One could view this as a "chain complex" of nonabelian groups, and the homology groups have a nice interpretation: they're the homotopy groups of the geometric realization |G|.

However, this method of passing to a graded object isn't in any sense an equivalence of categories. Moreover, it is known that simplicial groups mod weak equivalence are a model for the homotopy category of pointed CW-complexes. In some ways, then, we don't expect much in the way of purely algebraic characterizations of simplicial groups.

For simplicial rings there is the slight issue that the normalized chain functor plays somewhat poorly together with the tensor product, and so the Dold-Kan correspondence doesn't preserve the symmetric monoidal structure. This is not as serious an issue for associative algebras or commutative algebras in characteristic zero if one is only working up to weak equivalence/quasi-isomorphism, but if you are interested in an actual equivalence of categories it means that a simplicial commutative $mathbb{Q}$-algebra is not quite the same as a nonnegatively graded commutative differential graded algebra.