UPD: I didn't notice that you're asking about finite dimensional local systems, so this answer doesn't really answer your question. The easy way to fix that is to consider the category $D^b_f(Loc(X))$ of complexes of representations of $\pi_1$ with finite dimensional cohomology; in order to treat the case of an honest category of finite dimensional representations, it seems, the use of (derived) algebraic completion is unavoidable.
I think that both statements are true if $X$ is sufficiently nice and $K(\pi, 1)$, and if you consider bounded derived categories. Sufficiently nice here means that Exts between two local systems in the category of local systems and in the category of sheaves coinside with each other. I think it is enough to assume that $X$ is locally contractible; from this assumption follows the existence of the universal cover $\tilde X$.
The cone of two bounded complexes with locally constant cohomology has locally constant cohomology, because on a locally contractible space $X$ the category of locally constant sheaves is a so called Serre subcategory of the category of sheaves --- abelian subcategory closed under extensions. Then the long exac sequence of cohomology objects shows you that a cone of two complexes with locally constant cohomology has locally constant cohomology as well.
There are at least two ways to show that Exts between two local systems in two categories are the same. One can show that there are enough local systems, that are projective (resp., injective) as local systems, which are adapted to the functor $\mathrm{Ext}^*(-, V)$ (resp. $\mathrm{Ext}^*(V, -)$), where $V$ is also a local system. For projective local systems we can just take the regular representation of $\pi_1(X)$, let's call it $P$. For an injective, one can take the representation $\prod_{g \in G} \mathbb{Q}g$, where $\mathbb{Q}$ is some injective hull of your base ring. The $G$-action here is by multiplying to $g^{-1}$ from the right. Let's denote this module by $Q$. I think, by using the rule $v \mapsto \prod_{g \in G} g\otimes gv$ one can embed any $G$-module $V$ into $Q \hat\otimes V$, that is, infinite product of $Q$ ($V$ in the tensor product is considered with trivial $G$-action).
Now, both $P$ and $Q$ come from $\tilde X$, that is, if $\pi: \tilde X \longrightarrow X$ is the universal cover, then $P = \pi_! \underline{\mathbb{Z}}_{\tilde X}$ (where $\mathbb{Z}$ is the base ring) and $Q = \pi_*\underline{\mathbb{Q}}_{\tilde X}$. Since $\pi$ is a covering, there are no higher derived functors of $\pi_*$ and $\pi_!$.
Now one can use the adjointness $\mathrm{Ext}_{X}(F, Q) = \mathrm{Ext}_{\tilde X}(\pi^*F, \underline{\mathbb{Q}}_{\tilde X})$. Since $\mathbb{Q}$ is injective, $\tilde X$ is contractible and $\pi^*F$ is a locally constant (hence constant) sheaf, this vanishes in higher degrees. Or, you can show that $P$ is adapted by using the fact that for a covering map $\pi_!$ is left adjoint to $\pi^*$ (a fact which I think I know how to prove, but I was unable to find a reference for it in the case of an infinite covering. You also need to assume $X$ to be locally compact to use this, i suppose).
Now, using the induction on the length of a complex, you can show that the inclusion functor from $D^b(Loc(X))$ into $D^b(X)$ is a fully faithful embedding.
And since local systems generate $D^b_{Loc}(X)$ as a triangulated subcategory of $D^b(X)$, it is precisely the essential image of this functor. This is more or less standart; i think that an appendix to this Positselski's paper has a good treatment of that stuff.
Sadly, I don't know what will happen if one wants to consider unbounded categories.
