Model categories and chain complexes

I'm fairly new to thinking about homological algebra and chain complexes in their own right, i.e outside of isolated examples such as for constructing simplicial homology, or for computing $$Ext$$ groups for some Hopf algebroid.

Given an abelian category $$\mathscr{A}$$ with a category of chain complexes $$Ch(\mathscr{A})$$, the homotopy category $$K(\mathscr{A})$$ is defined as the naive homotopy category (i.e. replace chain maps with chain homotopy classes of chain maps) but with quasi-isomorphisms inverted (these are the maps which induce isomorphisms on homology). This is obviously the result of placing some kind of model structure on $$Ch(\mathscr{A})$$. This leads me to consider a few obvious questions.

1. Are there alternate, interesting model structures for $$Ch(\mathscr{A})$$?
2. (How) has the advancement of model category theory impacted the study of things such as chain complexes, and perverse sheaves for example?
3. Are there any good treatments of homological algebra which makes use of the rich theory of model categories?
• Note that there are in fact two model structures that are commonly used - the injective and projective model structures, which correspond to taking injective and projective resolutions respectively. (At least naively, each of these require different boundedness conditions on the chain complexes because we resolve in different directions. Putting a model structure on unbounded chain complexes is trickier. This should not be surprising because we cannot inductively construct injective/projective resolutions for unbounded chain complexes) – Dexter Chua May 8 at 9:19
• @DexterChua of course! When I say "alternate" model structures I certainly mean alternate to (at least) these two model structures. Could you say anything about the unbounded case? For example I'm guessing it still has weak equivalences the quasi-isomorphisms and I would certainly hope that it is cofibrantly generated – asd May 8 at 11:26