I think this question is a good one, but don't expect an encyclopedic answer — MO is not an encyclopedia. Here are some answers, with the disclaimer that I'm a category theorist but not an algebraic geometer.
To question A, by and large the 21st perspective will probably say that it is definitely a stack, in some notion of the word. Certainly there are applications where you do want to consider the "space" quotient $X/G$, in which points in the same orbit are honestly identified. This is like taking a form of "$\pi_0$" of the stack. (Not etale $\pi_0$, certainly, but a form of $\pi_0$ that's valued in spaces rather than sets.)
To questions B, D, and E, the answer is that, as you guessed, the best definition of $\operatorname{QCoh}(X/G)$ is the category of $G$-modules in $\operatorname{QCoh}(X)$, at least when $G$ is a finite group. The geometric intuition is that a quasicoherent sheaf on $X$ is something like a vector bundle over $X$. In the quotient $X/G$, we add an isomorphism between any two points for each way that they are related by an element of $G$. So a vector bundle over $X/G$ should have a fiber over each point of $X$, and an isomorphism between these fibers for each pair of $G$-related points.
There is a quotient morphism $X \to X/G$. The $\operatorname{QCoh}$ functor is best understood as contravariant, just like $\mathcal{O}$ is contravariant. Namely, a geometric morphisms $f: X \to Y$ correspond (modulo details) to symmetric monoidal "linear" functors $f^\ast : \operatorname{QCoh}(Y) \to \operatorname{QCoh}(X)$, which pull back a "vector bundle" along the map. This is certainly true for $\operatorname{QCoh}(X/G) = \operatorname{QCoh}(X)^G$, with the quotient morphism corresponding to the functor "forget the $G$-action". That said, each such functor $f^\ast$ also has a right adjoint $f_\ast : \operatorname{QCoh}(X) \to \operatorname{QCoh}(Y)$, which is not usually symmetric monoidal — it is that takes a "vector bundle" over $X$ and makes it into the "vector bundle" over $Y$ whose stock over $y\in Y$ is the space of sections over $f^{-1}(y)$ of the corresponding bundle on $X$. In the case of the quotient map $X \to X/G$, its right adjoint is the "free" functor, assigning to a quasicoherent module $M$ the corresponding free $G$-module $G \otimes M$. You ask for $f_\ast$ to be "a morphism of abelian categories", which is vague to me. The best definition I know of "morphism of abelian categories" is a right-exact functor (if I have left and right correct), in which case in general pushforward maps are not morphisms — they are instead left-exact. I think that if $G$ is finite, then in fact the pushforward along the quotient map is exact; maybe I need to include that the characteristic of the ground field does not divide the order of $G$.
As for C, as just a subcategory, I'm sure the answer is yes. If you ask for more conditions, the answer is probably still yes, at least in the finite-group case: you should be able to take the category of cocommutative coalgebras in $\operatorname{QCoh}(X/G)$ and find this as sheaves-of-sets on something. But I'd have to think more about details.
It is hard to know where to begin! A general principle is that as long as you are only concerned with the derived category of a single variety, it is generally sufficient to consider it as a triangulated category; while as soon as you are interested in families of derived categories, it becomes impossible to stay in the triangulated setting, so that it becomes necessary to pass to some homotopical refinement. The basic reason for this is the extremely poor behaviour of the homotopy theory of triangulated categories.
The most striking example is the work of Orlov on Fourier-Mukai functors. A triangulated functor between derived categories of schemes is called of Fourier-Mukai type if it of geometric origin, i.e. if it is given by twisting by a complex on the product of the schemes. Orlov showed in the 90's that for smooth projective varieties over a field, any fully faithful functor between derived categories of coherent sheaves is of Fourier-Mukai type. It was expected, or hoped, that every functor is Fourier-Mukai, but this turned out to be impossible to prove in the setting of triangulated categories, literally: a counter-example was finally found last year by Van den Bergh and Rizzardo (arXiv:1410.4039), while the dg-categorical analogue was proved ten years ago by Bertrand To\"en (arXiv:0408337). A brief glance at the latter paper will be enough to convince oneself that the proof relies heavily on studying the homotopy theory of dg-categories.
A second important application is algebraic K-theory. At the triangulated level, the only part of the algebraic K-theory of $X$ that can be defined from only the triangulated structure is the Grothendieck group. However, it has been believed that the algebraic K-theory of a scheme can be recovered from its derived category for a long time (in the sense that, if two schemes have triangulated-equivalent derived categories, then their higher K-theory groups are isomorphic). Neeman spent many years working on this problem, and I believe that in the end, his work implies that this is true under some assumptions like regularity. However, the proof takes up eight papers of around one hundred pages each. On the other hand, after work of Schlichting, To\"en and others, it is possible to define the algebraic K-theory of a dg-category in such a way that it recovers Thomason-Trobaugh K-theory for the dg-derived category of a scheme. In fact, one can deduce from this the analogous result for triangulated categories: using the theory of Fourier-Mukai functors mentioned above, it is possible to prove that the derived categories of two schemes (quasi-compact, separated over a field, if I recall correctly) are triangulated-equivalent iff their dg-derived categories are quasi-equivalent.
Related to this is the question of descent. As bananastack mentioned, the derived category satisfies Zariski descent only at the dg-level. In fact, at the dg-level it even satisfies Nisnevich descent. I believe the latter fact is due independently to Bhatt, Drinfeld and Lurie. From this it is possible to deduce a nice proof of the theorem of Thomason-Trobaugh on Nisnevich descent for K-theory. The missing ingredient is the localization fibre sequence, which can be deduced from compact generation properties of the derived category, which can in turn be proved in an elegant way at the dg-level, as explained in the beautiful paper arXiv:1002.2599 of To\"en.
Finally, let me note that, even though the various dg-enhancements constructed in the literature seem rather complicated, the dg-categorical derived category is in fact a more natural object than the triangulated one, in the following sense. The reason one considers the derived category of $X$ in the first place is in order to talk about "homotopy theory of chain complexes of coherent sheaves on $X$, up to quasi-isomorphism". The naive way to make this precise is to take the Gabriel-Zisman localization of the category of chain complexes at the class of quasi-isomorphisms, and then manually identify the distinguished triangles. The smart way is to take the Dwyer-Kan simplicial localization instead, which is a homotopically correct version of Gabriel-Zisman localization which doesn't kill higher homotopies; more precisely, it is a simplicially enriched category whose simplicial Hom-sets have $\pi_0$ identified with the Hom-sets of the Gabriel-Zisman localization. This gives rise to a dg-enhancement where the distinguished triangles are already built in as the (co)fibre sequences.
Best Answer
(Hopefully t3uji, tony pantev or Greg Stevenson will chime in with a more authoritative answer, but in the meanwhile..)
The notion of singular support of a coherent sheaf is an analog of the notion of singular support of a constructible sheaf or D-module. Let's quickly recall the latter: given a sheaf we can measure its failure to be locally constant at a particular point in a particular codirection. Namely given a covector at a point, i.e., a hyperplane in the tangent space, you ask if the sheaf behaves locally constantly moving off this hypersurface (this is a generalization of the Cauchy-Kovalevski theorem in PDE). You can measure this by taking a local function with the given covector as its differential at our point and calculating relative cohomologies (Morse groups) of its level sets near this point and "seeing if anything happens". The singular support (or microlocal support) is the collection of all covectors where our sheaf is not locally constant - ie all points and directions where "something interesting happens".
A very nice recent idea of several people (Isik, Arinkin-Gaitsgory, and others following on Orlov's work on categories of singularities --- someone who knows the history better please correct) is that one can do a very similar operation for coherent sheaves.
Recall that on a smooth variety any coherent sheaf (or bounded coherent complex) is quasiisomorphic to a perfect complex (bounded complex of vector bundles). This fails precisely at singular points of varieites by a theorem of Serre. Orlov introduced the category of singularities of a variety as the quotient of the bounded derived category by perfect complexes --- i.e., a measure of "how and where" the variety is singular (this category of singularities is supported at the singular locus). This is intimately related to the theory of matrix factorizations.
The new notion of singular support is an "individual" version of this construction for lci schemes (schemes with cotangent complex in degrees -1,0) - i.e. one looks at a specific coherent sheaf (or bounded complex) and attaches to it its "microlocal support" --- naively speaking, the collection of points and (degree -1) codirections where the sheaf fails to be perfect (see section 0.3.7 for a way to make this intuition precise, using a description of our scheme as a local complete intersection). One can also define the singular support as an honest support for the sheaf, when considered as a module for the Hochschild cohomology sheaf of the variety (self-Ext of the identity). One can also think roughly about representing covectors as differentials of functions, and then passing to categories of matrix factorizations of this function and seeing whether our sheaf survives - again, whether the sheaf is "interesting" at a given point and codirection.
Edit: Let me add a little about the role of Hochschild cohomology. By (one) definition the Hochschild cohomology of a scheme is the self-Ext of the identity functor of the (dg) derived category. In other words, the Hochschild cohomology tautologically acts by endomorphisms of every sheaf, in a way compatible with all morphisms. One can say more -- the Hochschild cohomology can be identified with the enveloping algebra of a Lie algebra structure on the shifted tangent complex $T[-1]$, which acts on every object via the construction of Atiyah classes (see Kapranov's paper on Rozansky-Witten theory for example, Markarian's preprint on HH and many more recent papers, the latest word maybe Calaque-van den Bergh). This can be nicely interpreted in terms of derived loop spaces -- $T[-1]$ is the Lie algebra of the free loop space, and Hochschild cohomology is its "group algebra"..
In our case we are interested in an lci scheme, and in particular the action of the top (+1) piece of the tangent complex, which after shift to $T[-1]$ lives in degree 2, or after taking enveloping algebra lives in even degree Hochschild cohomology. The singular support is then the support of the action of this commutative algebra (piece of even $HH^*$) on the sheaf. In other words, any sheaf on $X$ tautologically has a bigger action of a not quite commutative algebra, the Hochschild cohomology, but there's a commutative piece we can single out in there in the lci case and then take usual support.