It seems to me that everything written so far is addressing the title, not the body of the question, in particular focusing on the unstable. Part of that is people asking their own questions. But I think it is best to start with the easiest questions. Indeed, Tom, you have singled out the right place to start: Yes, the Atiyah-Hirzebruch spectral sequence is a hypercohomology spectral sequence.
Of course, we need a generalization or abstraction of the hypercohomology spectral sequence to make sense of that. Surely, you know that as you move from homological algebra to stable homotopy theory, you should generalize from derived categories of abelian categories to the abstraction of triangulated categories.* But there is another abstraction that you should know about, that of t-structures.
As Dylan says, spectral sequences come from filtrations.** But where do filtrations come from? One source is taking a chain complex and filtering by degree. The subquotients reveal exactly the original chain complex, while we'd prefer something that depended only on the quasi-isomorphism class. (Though often this is good enough because, although its $E_1$ page is not canonical, the rest of the sequence is.) The steps of this filtration are called the "naive truncations." They have the property that their homology agrees with the original in low degrees, is zero in high degrees, and is not canonical in a single degree. With a little modification, this can be changed into the good truncation, which has no such intermediate degree, but goes directly from agreeing to disagreeing.
Here is one version of the hypercohomology spectral sequence, I think pretty close to way that you see it. Start with a right-derivable functor $F\colon A\to B$ between abelian categories and the goal of understanding its derived functor on chain complexes $RF\colon DA\to DB$. Start with a complex $C\in DA$, take its good filtration, so that its subquotients are $H^iC[i]$, shifts of objects of $A$, and apply $RF$ to get a chain complex made of $R^jF(H^iC)$. Then as you reassemble $C$ from the $H^iC$, the spectral sequence reassembles $RF(C)$ from $RF(H^iC)$.
Let us abstract what this argument required: not that $DA$ was the derived category of an abelian category, but merely that each object had a filtration whose subquotients lived in shifts of an abelian category, called the heart; and that it was easier to understand the functor restricted to this abelian category. The first condition, which does not mention the functor, is called a t-structure.
Ultimately, the point is that the stable homotopy category has a t-structure whose heart is the category of abelian groups, but exotic t-structures abound and the concept was introduced in algebraic geometry. The easiest example is given by duality: The derived category of perfect chain complexes of abelian groups (that is, bounded complexes of finitely generated free abelian groups) is equivalent to its opposite under the contravariant duality functor $\mathrm{Hom}(-,\mathbb Z)$. Thus, we can transport the t-structure across the duality to get a new t-structure on the old category. The old heart is the category of finitely generated abelian groups (placed in degree zero). The new heart is equivalent to the opposite of the old heart. It consists of objects that are the sum of a free abelian group in degree zero and a torsion group shifted by $1$.
The filtration from the t-structure on the stable homotopy category is called the Postnikov filtration. The objects of its heart are called the Eilenberg-MacLane spectra; their associated cohomology theories are usual cohomology. Thus if our functor of interest is cohomology of a fixed space $X$ with coefficients in the varying spectrum $E$, its restriction to the heart is ordinary cohomology, a well-understood starting point, and so the hypercohomology spectral sequence is $H^i(X;E^j)\Rightarrow E^{i+j}(X)$, the same form as the Atiyah-Hirzebruch spectral sequence. (You may have to do some work to check that it is the actually the same spectral sequence.)
* Probably what I have to say can be made to work under Verdier's axioms for triangulated categories, but I really mean stable $\infty$-categories. Or you could work with DG-categories, until you want to move to stable homotopy theory.
** The Bockstein spectral sequence is a purely algebraic spectral sequence that I do not know how to see as coming from a filtration, though I have a vague memory of another spectral sequence that does come from a filtration and carries the same information.
Best Answer
Cosheaves are indeed mysterious gadgets. On the one hand, cosheaves are everywhere, but on the other hand, someone used to thinking sheaf-theoretically may have some problems. I am very close to finishing an exposition on cosheaves, but need another week or so to put it on the arxiv. Bredon's book on sheaf theory has the most complete reference on cosheaves, so you might look there if you like.
AS you may know, pre-cosheaves are just covariant functors $\hat{F}:\mathrm{Open}(X)\to\mathcal{D}$ where $\mathcal{D}$ is some "data category" like Vect, Ab, or what have you. Cosheaves send covers (closed under intersection) to colimits and different covers of the same open set get sent to isomorphic colimits. The Mayer-Vietoris axiom is a good way of thinking about cosheaves and since homology commutes with direct limits, one can see that $H_0(-,k)$ is always a cosheaf. In particular, $H_0(-,\mathcal{L})$ is a cosheaf whenever $\mathcal{L}$ is a local system.
As you observed, since cosheaves are fundamentally colimit-y, they have left-derived functors rather than right-derived ones. Thus the answer to (1) is yes.
In regards to (2), one must be careful. I believe the answer is yes, but allow me to pontificate on the problem.
Filtered limits and finite colimits do not commute in most categories like Ab, Vect, or Set. This has serious ramifications through the theory of cosheaves.
For example, it is not necessarily true that a sequence of cosheaves is exact iff it is exact on costalks. Here costalks are defined using (filtered) inverse limits rather than direct ones.
Another very serious consequence is that Grothendieck's sheafification procedure cannot be dualized to give cosheafification. Thus the usual phrase
"let blah by the cosheaf associated to the pre-cosheaf blah"
is not necessarily well-founded because it is unclear how to cosheafify! People have solved this problem in the past by working with pro-objects (which corrects for this "filtered limits not commuting with finite colimits" asymmetry) and then they use Grothendieck's construction. However, for abstract categorical reasons one can check that cosheafification does exist for data categories like Vect (i have worked out a proof and haven't found in the literature anyone who claims to have proved this), we just don't have an explicit construction. That said, the usual description of the left-derived functor of the push-forward should still hold.
On the other hand, if one works in the constructible setting, one can get the statements you would like. In particular, it is true that cosheaves constructible with respect to a cell structure are derived equivalent to sheaves constructible with respect to the same cell structure. I discovered independently my own proof, only to find that at least two other people have proved this before. However, in my opinion, the equivalence is the "correct" form of Verdier duality. A larger and updated exposition should be available soon.