I've found the following somewhat intricate way of answering Q1 in the affirmative. Any complex in $D(Ab)$ quasi-isomorphic to a graded abelian group. Hence, it is enough to consider complexes concentrated in a single degree. Given an abelian group $A$ and $n\in\mathbb Z$, let $(A,n)$ be the abelian group $A$ concentrated in degree $n$. For simplicity, I will use the same notation for the Eilenberg-MacLane spectrum $\Sigma^nHA$. In the derived category we have,
$$D(Ab)((A,n),(B,n))=\operatorname{Hom}(A,B),$$
$$D(Ab)((A,n),(B,n+1))=\operatorname{Ext}(A,B),$$
$$D(Ab)((A,n),(B,m))=0\text{ otherwise}.$$
In the stable homotopy category we have the stable Eilenberg-MacLane groups
$$SH((A,n),(B,m))=H^{m+k}(A,n+k;B),\quad k>>0.$$
It is well known, since E-ML's "On the groups..." (Annals) that
$$SH((A,n),(B,n))=\operatorname{Hom}(A,B),$$
$$SH((A,n),(B,n+1))=\operatorname{Ext}(A,B),$$
and that the functor $D(Ab)\rightarrow SH$ is the identity on the previous morphism sets. Hence we are done. The groups $SH((A,n),(B,m))$ are however non-trivial for $m>n+1$, in general.