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.)
I am not knowledgeable enough to have much to say I have not writen in my answer to a previous question of yours, and I think that David Roberts's answer (or, rather immodestly, my previous one) provides what you were looking for as regards your first question. Just a few additional small points:
Pursuing Stacks is not a letter. See Tim Porter's comment.
As regards Grothendieck's opinion of Thomason's model structure, I do not know. Actually, I am unsure he knew of Thomason's model structure when writing Pursuing Stacks [EDIT: see Tim Porter's comment below]. What he knew for sure was that the localization of $Cat$ with respect to classical weak equivalences (functors between small categories the nerve of which are simplicial weak equivalences) is equivalent to the classical homotopy category. The first proof is due to Quillen and Illusie "wrote the details" (his words) in his thesis. (And there is a quite simpler proof, by the way.) Model structures crop up in Pursuing Stacks at some point, but I am pretty sure the idea is not developed in the beginning, which is much more concerned with mere models for homotopy types. Here is a citation from Chapter 75: "the notion of asphericity structure — which, together with the closely related notion of contractibility structure, tentatively dealt with before, and the various "test notions" (e.g. test categories and test functors) seems to me the main payoff so far of our effort to come to a grasp of a general formalism of "homotopy models"." (Beware: these asphericity structures are not what Maltsiniotis called "asphericity structures" in his own work.)
Another fact Grothendieck knew was, of course, Quillen's Theorem A. It seems he did not write a detailed proof of the relative version, but he gave a sketch of a toposic proof of it, though, and took it as an axiom for what he called basic localizer.
As for your second question, I do not know, but it seems to me that Grothendieck was not that interested in simplicial sets and thus did not work extensively with them. In a 1991 letter to Thomason, he wrote: " D’autre part, pour moi le "paradis originel" pour l’algèbre topologique n’est nullement la sempiternelle catégorie ∆∧ semi-simpliciale, si utile soit-elle, et encore moins celle des espaces topologiques (qui l’une et l’autre s’envoient dans la 2-catégorie des topos, qui en est comme une enveloppe commune), mais bien la catégorie Cat des petites catégories, vue avec un œil de géomètre par l’ensemble d’intuition, étonnamment riche, provenant des topos. En effet, les topos ayant comme catégories des faisceaux d’ensembles les C∧ , avec C dans Cat, sont de loin les plus simples des topos connus, et c’est pour l’avoir senti que j’insiste tant sur l’exemple de ces topos ("catégoriques") dans SGA 4 IV". (See here.)
To conclude, let me mention that, if one takes Grothendieck's viewpoint of homotopical algebra, there should exist not only a homotopy theory of categories, but a homotopy theory of $n$-categories. In this respect, there should be a "relative Theorem A" for every $n$, which should allow one to define a workable notion of "basic $n$-localizer". (Actually, this is already done for $n=2$: see this paper by Bullejos and Cegarra for Theorem A.) And then one should work out a theory of test $n$-categories, whose $(n-1)-Cat$-valued presheaves should be models for homotopy types, and so on. To sum up, what Grothendieck wanted to do amounts to giving new foundations for homotopical algebra, and this is still a work in progress.
David Roberts gives the two most useful available references in his answer. If you want to read Grothendieck's words (and in English), just wait for the upcoming annotated version of Pursuing Stacks.
EDIT (2013/10/29): Rereading this answer, I realize that I should add something of which I was not aware at the time of my writing, still regarding Grothendieck's knowledge of Thomason's model category structure (see also Tim Porter's comment and David Roberts's answer). An annotated version of section 69 of Pursuing Stacks is available at http://www.math.jussieu.fr/~maltsin/groth/ps/ps-69.pdf. On page 4, Grothendieck writes that "it appears very doubtful still that (Cat) is a “model category” in Quillen’s sense, in any reasonable way (with W of course as the set of “weak equivalences”". Thus, he was not aware of the existence of Thomason's structure then. See also note 6 on that same page: Grothendieck has learnt of the existence of Thomason's model structure between the writing of Sections 69 and 87.
Best Answer
As you say, a major use of $(\infty,2)$-categories is for organising $(\infty,1)$-categories and similar objects (stable $\infty$-categories, $\infty$-topoi, enriched $\infty$-categories, $\infty$-operads...). The importance of the non-invertible 2-cells is the same as in classical $(2,2)$-category theory: they provide natural internal notions of adjunction, base change mappings, lax functor, lax monoidal functor, Kan extension, and so on. An $(\infty,2)$-category can be used to organise collections of these structures and keep track of coherences between them.
A good illustration of the utility of this is the notion of a "six-functor formalism," which Gaitsgory and Rozenblyum (https://bookstore.ams.org/surv-221/) argued is best captured by a certain symmetric monoidal $(\infty,2)$-functor on an $(\infty,2)$-category of correspondences between derived stacks. We definitely need $\infty$ here because the value of such a functor would be something like the derived $\infty$-category of quasi-coherent or constructible sheaves, and the source may also include some derived/higher objects. We definitely need $2$ because the 2-cells of the category of correspondences encode all kinds of coherences between the six functors (for example, base change 2-cells and the higher associativity of compositions of 2-d grids of base change squares). Even if one ultimately only cares about constructing functors out of correspondences on a 1-category, in practice one still needs its universal property among $(\infty,2)$-categories.
See also my paper https://arxiv.org/abs/2005.10496 for a slightly different take to Gaitsgory-Rozenblyum's.