I am trying to learn about simplicial commutative rings, and would be grateful if one can help with some basic facts about them. Basically, I would like to understand how to do homological algebra over a simplicial ring.


2. Let $A$ be a simplicial commutative ring. 
Is the category of simplicial modules over $A$ abelian?
3. If so, does its derived category exist?
4. Does any simplicial module have a projective resolution? a flat resolution? an injective resolution?
5. Consider the associated DG-algebra $N(A)$. Is there any relation between the derived 
category of DG-modules and the derived category I was hoping exists in 3 above?
6. Any references that discuss these basic issues?



Basic questions about simplicial commutative rings
Yes, there are many references that discuss this. The first is Quillen's *Homotopical Algebra*. Chapter II contains much of what you're asking about, especially II.4 and II.6


For a more modern version, see Schwede-Shipley *Algebras and Modules in Monoidal Model Categories*. Section 5 is all examples and contains precise references in Quillen's work above.


This covers your (2), (3), and (6). I agree with Zhen Lin that (4) is the wrong question to ask, since the structure as an abelian category is less important than the model structure if you are trying to study the homotopy category from (3). For (5), check out the Dold-Kan theorem. A nice write-up is in Goerss-Schemmerhorn *Model Categories and Simplicial Methods* 4.1. This also includes the model structure on simplicial R modules and what resolutions look like for simplicial R modules, answering your (4) in the correct homotopically meaningful sense. It is clear from 4.1 that $N$ is both an equivalence of categories (if by DG we mean non-negatively graded) and a Quillen equivalence, so the homotopy theories agree, and this answers your (5)


