Mostly I refer you to my answer here and also this question.

To answer the question about (co)fibrations: No, there is no notion corresponding to (co)fibration in the (∞,1)-category associated to a model category. After all, being a (co)fibration has no homotopical information: every map is equivalent to a (co)fibration. For the sorts of things you need the (co)fibrations to define in model categories, such as homotopy (co)limits, you can give direct definitions in terms of mapping spaces in the (∞,1)-category.

There are two sensible notions of "sameness" of model categories: categorical equivalence, by which I mean an equivalence of categories which preserves each of the three classes of arrows, and Quillen equivalence. This is a lot like the difference between two objects in a model category being isomorphic or merely weakly equivalent, though I don't think anyone has a framework in which to make this idea precise. When you consider, say, the projective and injective model structures on a diagram category, these model structures are Quillen equivalent but not categorically equivalent. They have different 1-categorical properties (it's easy to describe left Qullen functors out of the projective model structure and left Quillen functors into the injective model structure) but they model the same homotopy theory. The passage to associated (∞,1)-categories eliminates the distinction between categorical equivalence and Quillen equivalence: two model categories are Quillen equivalent if and only if their associated (∞,1)-categories are equivalent. (Actually, I am not sure whether there are some technical conditions needed for the last assertion, but if so they are satisfied in practice.)

Mostly I refer you to my answer here and also this question.

To answer the question about (co)fibrations: No, there is no notion corresponding to (co)fibration in the (∞,1)-category associated to a model category. After all, being a (co)fibration has no homotopical information: every map is equivalent to a (co)fibration. For the sorts of things you need the (co)fibrations to define in model categories, such as homotopy (co)limits, you can give direct definitions in terms of mapping spaces in the (∞,1)-category.

There are two sensible notions of "sameness" of model categories: categorical equivalence, by which I mean an equivalence of categories which preserves each of the three classes of arrows, and Quillen equivalence. This is a lot like the difference between two objects in a model category being isomorphic or merely weakly equivalent, though I don't think anyone has a framework in which to make this idea precise. When you consider, say, the projective and injective model structures on a diagram category, these model structures are Quillen equivalent but not categorically equivalent. They have different 1-categorical properties (it's easy to describe left Qullen functors out of the projective model structure and left Quillen functors into the injective model structure) but they model the same homotopy theory. The passage to associated (∞,1)-categories eliminates the distinction between categorical equivalence and Quillen equivalence: two model categories are Quillen equivalent if and only if their associated (∞,1)-categories are equivalent. (Actually, I am not sure whether there are some technical conditions needed for the last assertion, but if so they are satisfied in practice.)

Mostly I refer you to my answer here and also this question.

To answer the question about (co)fibrations: No, there is no notion corresponding to (co)fibration in the (∞,1)-category associated to a model category. After all, being a (co)fibration has no homotopical information: every map is equivalent to a (co)fibration. For the sorts of things you need the (co)fibrations to define in model categories, such as homotopy (co)limits, you can give direct definitions in terms of mapping spaces in the (∞,1)-category.

Mostly I refer you to my answer here and also this question.

To answer the question about (co)fibrations: No, there is no notion corresponding to (co)fibration in the (∞,1)-category associated to a model category. After all, being a (co)fibration has no homotopical information: every map is equivalent to a (co)fibration. For the sorts of things you need the (co)fibrations to define in model categories, such as homotopy (co)limits, you can give direct definitions in terms of mapping spaces in the (∞,1)-category.

There are two sensible notions of "sameness" of model categories: categorical equivalence, by which I mean an equivalence of categories which preserves each of the three classes of arrows, and Quillen equivalence. This is a lot like the difference between two objects in a model category being isomorphic or merely weakly equivalent, though I don't think anyone has a framework in which to make this idea precise. When you consider, say, the projective and injective model structures on a diagram category, these model structures are Quillen equivalent but not categorically equivalent. They have different 1-categorical properties (it's easy to describe left Qullen functors out of the projective model structure and left Quillen functors into the injective model structure) but they model the same homotopy theory. The passage to associated (∞,1)-categories eliminates the distinction between categorical equivalence and Quillen equivalence: two model categories are Quillen equivalent if and only if their associated (∞,1)-categories are equivalent. (Actually, I am not sure whether there are some technical conditions needed for the last assertion, but if so they are satisfied in practice.)