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.
The key obervation is that if you have a map $\phi\colon A\to B$, then you can choose $M_A$ and $M_B$ so that this extends to $\tilde{\phi}\colon M_A\to M_B$: choose any $M_A$ you like, and then choose $M_B=M_{M_A\oplus B}/A$ where we embed $A$ in $M_A\oplus B$ by $(a,-\phi(a))$. The map $\tilde{\phi}\colon M_A\to M_B$ and inclusion of $B$ are induced by the obvious maps to $M_A\oplus B$. This allows us to transfer naturality from the lower degree.
For the connecting maps, I believe you can show that $M_B/M_A$ can be chosen as $M_C$ (look at the long exact sequence). By the 9-lemma, we also have a SES given by $0\to M_A/A\to M_B/B \to M_C/C\to 0$; commutation with the boundary map a degree down for this sequence gives it for $0\to A\to B \to C\to 0$. I don't blame Grothendieck for not writing this out; it hurts my head to try to picture the diagrams.
Best Answer
Grothendieck's Tohoku paper was an attempt to set the foundations of algebraic topology on a uniform basis, essentially to describe a setting where one can do homological algebra in a way that makes sense. He did this by using the concept of abelian categories. Perhaps a better question to ask yourself is "Why are abelian categories a good idea?" In answering your question, I will do some major handwaving and sacrifice some rigor for the sake of clarity and brevity, but will try to place the Tohoku paper in context.
At the time, the state-of-the-art in homological algebra was relatively primitive. Cartan and Eilenberg had only defined functors over modules. There were some clear parallels with sheaf cohomology that could not be mere coincidence, and there was a lot of evidence that their techniques worked in more general cases. However, in order to generalize the methodology from modules, we needed the category in question to have some notion of an exact sequence. This is a lot trickier than it might seem. There were many solid attempts to do so, and the Tohoku paper was a giant step forward in the right direction.
In a nutshell, Grothendieck was motivated by the idea that $Sh(X)$, the category of sheaves of abelian groups on a topological space $X$ was an abelian category with enough injectives, so sheaf cohomology could be defined as a right-derived functor of the functor of global sections. Running with this concept, he set up his famous axioms for what an abelian category might satisfy.
Using the framework given by these axioms, Grothendieck was able to generalize Cartan and Eilenberg's techniques on derived functors, introducing ideas like $\delta$-functors and $T$-acyclic objects in the process. He also introduces an important computational tool, what is now often called the Grothendieck spectral sequence. This turns out to generalize many of the then-known spectral sequences, providing indisputable evidence that abelian categories are the "right" setting in which one can do homological algebra.
However, even with this powerful new context, there were many components missing. For instance, one couldn't even chase diagrams in general abelian categories using the techniques from Tohoku in and of itself, because it did not establish that the objects that you wanted to chase even existed. It wasn't until we had results like the Freyd-Mitchell embedding theorem that useful techniques like diagram chasing in abelian categories became well-defined. Henceforth, one had a relatively mature theory of homological algebra in the context of abelian categories, successfully generalizing the previous methods in homological algebra. In other words, we have "re-interpreted the basics of [algebraic] topology" by allowing ourselves to work with the more general concept of abelian categories.