DG categories in algebraic geometry - guide to the literature?
Let me try to address the bulleted questions and simultaneously advertise the G-R book everyone has mentioned. Since the main question was about literature, I could also mention Drinfeld's article "DG quotients of DG categories," which nicely summarizes the state of the general theory before $\infty$-categories shook everything up. However, it doesn't contain any algebraic geometry.
If $X = \text{Spec } A$ is an affine scheme, it's reasonable to define the category of quasicoherent sheaves $\text{QCoh}(X) := A\text{-mod}$ as the category of $A$-modules. Any other definition (e.g. via Zariski sheaves) must reproduce this answer anyway. If we understand this as the derived category of $A$-modules, then there is a canonical DG model: the homotopically projective complexes in the sense of Drinfeld's article.
The next step is to construct $\text{QCoh}(X)$ for $X$ not necessarily affine. So write $X = \cup_i \text{Spec } A_i$ as a union of open affines (say $X$ is separated to simplify things). It would be great if we could just "glue" the categories $A_i\text{-mod}$, the way that we compute global sections of a sheaf as a certain equalizer. Concretely, a complex of sheaves on $X$ should consist of complexes of $A_i$-modules for all $i$, identified on overlaps via isomorphisms satisfying cocycle "conditions" (really extra data). This is the kind of thing that totally fails in the triangulated world: limits of 1-categories just don't do the trick. Even if we work with the DG enhancements, DG categories do not form a DG category, so this doesn't help.
As you might have guessed, this is where $\infty$-categories come to the rescue. Let me gloss over details and just say that there is a (stable, $k$-linear) $\infty$-category attached to a DG category such as $A$-mod, called its DG nerve. If we take the aforementioned equalizer in the $\infty$-category of $\infty$-categories, then we do get the correct $\infty$-category $\text{QCoh}(X)$, in the sense that its homotopy category is the usual derived category of quasicoherent sheaves on $X$. (Edit: As Rune Haugseng explains in the comments, it's actually necessary to take the limit of the diagram of $\infty$-categories you get by applying $\text{QCoh}$ to the Cech nerve of the covering. The equalizer is a truncated version of this.)
But, you might be thinking, I could have just constructed a DG model for $\text{QCoh}(X)$ using injective complexes of Zariski sheaves or something. That's true, and obviously suffices for tons of applications, but as soon as you want to work with more general objects than schemes you're hosed. True, there are workarounds using DG categories for Artin stacks, but the theory gets very technical very fast.
If we instead accept the inevitability of $\infty$-categories, we can make the following bold construction. A prestack is an arbitrary functor from affine schemes to $\infty$-groupoids (i.e. spaces in the sense of homotopy theory). For example, affine schemes are representable prestacks, but prestacks also include arbitrary schemes and Artin stacks. Then for any prestack $\mathscr{X}$ we can define $\text{QCoh}(\mathscr{X})$ to be the limit of the $\infty$-categories $A\text{-mod}$ over the $\infty$-category of affine schemes $\text{Spec } A$ mapping to $\mathscr{X}$. A cofinality argument for Zariski atlases shows this agrees with our previous definition for $\mathscr{X}$ a scheme.
For example, if $\mathscr{X} = \text{pt}/G$ is the classifying stack of an algebraic group $G$, then the homotopy category of $\text{QCoh}(\mathscr{X})$ is the derived category of representations of $G$. Even cooler: if $X$ is a scheme the de Rham prestack $X_{\text{dR}}$ is defined by $$\text{Map}(S,X_{\text{dR}}) := \text{Map}(S_{\text{red}},X).$$ Then, at least if $k$ has characteristic zero, our definition of $\text{QCoh}(X_{\text{dR}})$ recovers the derived category of crystals on $X$, which can be identified with $\mathscr{D}$-modules. So we put two different ``flavors" of sheaf theory on an equal footing.
There are plenty of interesting dg-categories one can associate to a scheme. From the point of view of six functor yoga, these should be viewed as "categories of coefficients" for cohomology theories. For example, the derived category of quasi-coherent sheaves (or its various variants) is the category of coefficients for coherent cohomology, just as the derived category of $\ell$-adic sheaves is the category of coefficients for $\ell$-adic cohomology, or motivic complexes are the coefficients for motivic cohomology.
These have been studied for decades using the language of triangulated categories, but it is well-known that they can each be defined as dg-categories. As the base varies, they form stacks of dg-categories with good descent properties (depending on the category of coefficients). In fact, in each of these examples, there is much more structure: there is a whole six functor formalism, categorifying the standard features of cohomology theories, like Künneth formulas, Poincaré duality, Gysin maps, etc. The six functor formalisms also lift to the dg-level.
For any given category of coefficients you might be interested in, there are certainly plenty of references, though most of them will be written in the language of triangulated categories. You shouldn't be bothered by this: there is a very large amount of interesting algebraic geometry you can do in this language, and anyway the arguments can be translated to more modern language without changing very much how they look.
On the other hand, if you are specifically interested in seeing the power of the modern language, one important point is the failure of descent at the triangulated level. The book of Gaitsgory-Rozenblyum is a great place to see descent arguments in practice. Another very good reference is the work of Bhatt-Scholze.
I should note that both these references actually use the language of $(\infty,1)$-categories instead of dg-categories. In fact, Gaitsgory-Rozenblyum's definition of dg-category is just a $k$-linear stable $(\infty,1)$-category. If you work in the $(\infty,1)$-category of dg-categories, you don't see any difference between this and the classical definition. If you do want to work more set-theoretically, you will have to pay for the psychological comfort by doing a lot of extra work to make sure the constructions you do are homotopically meaningful (e.g. whenever you take a tensor product or internal hom). If that's your preference, then I would suggest papers of Kuznetsov, Lunts and Orlov (which definitely contain a lot of interesting and beautiful mathematics).
Besides Gaitsgory-Rozenblyum (http://www.math.harvard.edu/~gaitsgde/GL/), you might try looking at Lee Cohn's work (http://arxiv.org/abs/1308.2587), which establishes some equivalence between the approaches. Abstract follows:
Differential Graded Categories are k-linear Stable Infinity Categories
Lee Cohn (Submitted on 12 Aug 2013) We describe a comparison between pretriangulated differential graded categories and certain stable infinity categories. Specifically, we use a model category structure on differential graded categories over k (a field of characteristic 0) where the weak equivalences are the Morita equivalences, and where the fibrant objects are in particular pretriangulated differential graded categories. We show the underlying infinity category of this model category is equivalent to the infinity category of k-linear stable infinity categories.