Ends and coends should be thought of as very canonical constructions: as Finn said, they can be described as weighted limits and colimits, where the weights are hom-functors.
Recall that if $J$ is a (small) category, a weight on $J$ is a functor $W: J \to Set$. The limit of a functor $F: J \to C$ with respect to a weight $W$ is an object $lim_J F$ of $C$ that represents the functor
$$C^{op} \to Set: c \mapsto Nat(W, \hom_C(c, F-)).$$
Dually, given a weight $W: J^{op} \to Set$, the weighted colimit of $F: J \to C$ with respect to $W$ is an object $colim_J F$ that represents the functor
$$C \to Set: c \mapsto Nat(W, \hom_C(F-, c)).$$
Then, as Finn notes above, the end of a functor $F: J^{op} \times J \to C$ is the weighted limit of $F$ with respect to the weight $\hom_J: J^{op} \times J \to Set$, and the coend is the weighted colimit of $F$ with respect to $\hom_{J^{op}}: J \times J^{op} \to Set$.
The ordinary limit of $F$ is the weighted limit of $F$ with respect to the terminal functor $t: J \to Set$. Ordinary limits suffice for ordinary ($Set$-based) categories, but they are inadequate for enriched category theory. The concept of weight was introduced to give an adequate theory of enriched limits and colimits (replacing $Set$ by suitable $V$, and functors as above by enriched functors, etc.)
Weighted colimits and weighted limits (in particular coends and ends) can be expressed in terms of Kan extensions. For any weight $W$ in $Set^{J^{op}}$, the weighted colimit of $F: J \to C$ (if it exists) is the value of the left Kan extension of $F: J \to C$ along the Yoneda embedding $y: J \to Set^{J^{op}}$ when evaluated at $W$, in other words
$$(Lan_y F)(W)$$
A similar statement can be made for weighted limits, as values of a right Kan extension.
You cannot compare both cohomologies unless $H$ has degree $1$, because otherwise the cohomology of $d+H\wedge$ is not $\mathbb{Z}$-graded.
If $H$ has degree $1$ then the cohomology of $d+H\wedge$ is the cohomology of $X$ with local coefficients corresponding to the flat line bundle with $1$-form $H$.
The only twisted coefficients associated to singular cohomology with real coefficients are the usual local coefficiens. This follows from the fact that, for $E_n=K(\mathbb{R},n)$, the bundles over $X$ with fiber $E_n$ are classified by $B(\operatorname{Aut}^h(E_n))=K(\operatorname{Aut}\mathbb{R},1)$. Here $\operatorname{Aut}^h$ is the topological group (or $A$-infinity space) of self-homotopy equivalences, and $\operatorname{Aut}$ is just an automorphism group. Therefore, bundles over $X$ with fiber $E_n$ are classified by homomorphisms $\pi_1(X)\rightarrow \operatorname{Aut}(\mathbb{R})$, which correspond to local systems.
Best Answer
I doubt that someone could learn higher category theory (and more in general higher dimensional algebra) without first studying a little of category theory, mostly because the definition given in such context use a lot of category theoretic machinery. About the textbook reference: MacLane's "Category theory for working mathematicians" may be a little outdated but I think it is still one of the most complete book of basic category theory second just to Borceux's books. Anyway there isn't a best book to learn basic category theory, any person could find a book better than another one, so I suggest you to take a look a some of these books, then choose which one is the best for you:
S. MacLane: Category theory for working mathematicians (I've already said a lot about this)
S. Awodey: Category theory (Peculiar because it has very low prerequisites and it's rich of examples too)
J. Adamek,H. Herrlich, G. Strecker: Abstract and concrete category theory (freely avaible at at this site "http://katmat.math.uni-bremen.de/acc/acc.pdf", maybe the book with the greatest number of examples from topology and algebra)
After you have read one of these book, you could also use Borceux's books and read some more advanced chapter of category theory which aren't discussed in the previous books.
F. Borceux: Handbook of Categorical Algebra 1: Basic Category Theory
F. Borceux: Handbook of Categorical Algebra 2: Categories and Structures
F. Borceux: Handbook of Categorical Algebra 3: Categories of Sheaves
For higher category theory I know just few reference:
Leinster's "Higher Operads Higher Categories" (http://arxiv.org/abs/math/0305049),
and
Lurie's "Higher Topos Theory" (http://arxiv.org/abs/math/0608040)
other good reference in higher category theory and higher dimensional algebra in general are Baez'This week's finds and arxiv articles Higher dimensional algebra*.
Hope this may help.