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.