Model categories and chain complexes

I'll take your question as license to advertise a relatively recent paper in a slightly more specialized but concretely calculational direction: http://nyjm.albany.edu/j/2014/20-53p.pdf. Its title is Six model structures for DG-modules over DGAs: Model category theory in homological action. The theme is how different model structures can illuminate concrete calculations. I had computed the cohomology of various homogeneous spaces, way back in the 1960's, using some strange looking explicit cochain complexes. Tobi Barthel, Emily Riehl and I found that those turn out to be explicit examples of a variant kind of cofibrant approximation.

Yes, there are zillions of model structures on Ch(A), corresponding to whatever class of projectives you choose to use for your homological algebra. This is all spelled out in the paper "Quillen model structures for relative homological algebra" by Christensen and Hovey.

More generally, the theory of cotorsion pairs builds model structures in many algebraic settings (including quasi-coherent sheaves; I'm not sure about perverse sheaves, not even if they satisfy bicompleteness as a category). Hovey's seminal paper is here. A great survey by Gillespie is here. This material has also appeared in books. A recent one by Marco Perez is here.