Does every (co)homology functor (in particular, stable homotopy) factor through chain complexes? Ordinary homology and cohomology factor through chain complexes via singular homology and cohomology. What about other (co)homology theories?
That is, for each spectrum $E$, do we have a lift in the following diagram?
$\begin{array}[ccc]
& \mathsf{HoTop} & \overset{E}{\to} & \mathsf{GrAb} \\
& \underset{?}{\searrow} & \uparrow \\
& & \mathcal{D}(\mathsf{Ch}_{\mathbb{Z}})
\end{array}$
Where $\mathsf{HoTop}$ is the homotopy category, $E$ is $E$-homology or $E$-cohomology (in which case it's contravariant, of course), $\mathsf{GrAb}$ is graded abelian groups, $\mathcal{D}(\mathsf{Ch}_{\mathbb{Z}})$ is the derived category of chain complexes in abelian groups, the functor $\mathcal{D}(\mathsf{Ch}_{\mathbb{Z}}) \to \mathsf{GrAb}$ is homology, and the functor labeled "?" is the desired lifting.
It would just about suffice to do this in the universal example of stable homotopy ("just about" only because we have to do this for all spectra now), because we have a factorization:
$\begin{array}[ccc]
& \mathsf{HoTop} & \overset{E}{\to} & \mathsf{GrAb} \\
\downarrow & \underset{\pi^S}{\nearrow} \\
\mathsf{HoSp}
\end{array}$
where $\mathsf{HoSp}$ is the stable homotopy category, $\pi^S$ takes stable homotopy groups, and $\mathsf{HoTop} \to \mathsf{HoSp}$ is either $E\wedge(\Sigma^\infty-)$ (for homology) or $\operatorname{Fun}(\Sigma^\infty -,E)$ (for cohomology).
So in some sense I'm asking about the relationship between two sorts of graded-abelian-group-valued decategorification: taking homotopy groups of a spectrum, and taking homology groups of a chain complex.
 A: No (I mean, not in a triangulated way), otherwise any generalized homology theory of a mod 2 Moore space would be 2-torsion, but this is not true for mod 2 stable homotopy groups (it's well known that you get a cyclic group of order 4). For positive results under extra hypotheses see:
Heller, A., 1966. Extraordinary Homology and Chain Complexes, in:  Proceedings of the Conference on Categorical Algebra, La Jolla. pp. 355–365.

Neeman, A., 1992. Stable homotopy as a triangulated functor. Invent Math 109, 17–40. doi:10.1007/BF01232016

A: As written by Fernando, the answer is no if you suppose that you have exactness fo pairs at the chain level. In fact for a functor 
$$L_*:CW^{pairs}\rightarrow Ch(Ab)$$
if you have a short exact sequence of chain complexes
$$0\rightarrow L_*(A,\emptyset)\rightarrow L_*(X,\emptyset)\rightarrow L_*(X,A)\rightarrow 0$$
and if the homology $\mathcal{L_*}=H_*(L_*(-))$ is a generalized homology theory then we have:
Theorem (Burdick, Conner, Floyd 1968).To each finite CW-pair (X,A) we have a natural isomorphism:
$$\mathcal{L}_n(X,A)\cong \bigoplus_{p+q=n} H_p(X,A;\mathcal{L}_q(pt)).$$
Ref: "Chain Theories and their Derived Homologies", Proc . Amer. Math. Soc. l9 (l968) Proceedings of the AMS (1968)
