Edit & Note: I'm adopting a declaring a convention here because I don't feel like trying to fix this in a bunch of spots: If I said model category and it doesn't make sense, I meant a model-category "model" of an (infinity,1)-category.

At the very beginning of Lurie's higher topos theory, he mentions that there is a theory of $(\infty,1)$-categories that can be directly constructed by using model categories.

What I'd like to know is:

Where can I find related papers (Lurie mentions two books that are not available for free)?

How dependent is the theory developed in HTT on quasicategories? Can the important results be proven for these $(\infty,1)$-model-categories by proving some sort of equivalence (not equivalence of categories, but some weaker kind of equivalence) to the theory of quasicategories?

When would we want to use quasicategories rather than these more abstract model categories?

And also, conversely, when would we want to look at model categories rather than quasicategories?