This is just to answer your first question. The second one I don't know about.

The homotopy theory of categories is not quite as you envisage it. Really Grothendieck is thinking of the Thomason model structure on $Cat$ (the category of small categories), which is Quillen equivalent to the Quillen model structure on $sSet$ via the nerve functor. Then Grothedieck considered pairs $(Cat,W)$ where $W$ is a class of functors which acted as weak equivalences. This he called a _basic localizer_ (<a href="http://ncatlab.org/nlab/show/basic+localizer">nLab</a>). Grothendieck conjectured, and Cisinski proved, that the class of weak equivalences in the Thomason model structure was the smallest basic localizer.


From there Grothendieck moved to considering pairs $(C,W)$ for any category $C$ and class $W$ of arrows such that $C[W^{-1}]$ was equivalent to the homotopy category of CW-complexes, or even the homotopy category of some basic localizer, and in particular he was interested in when $C = Pre(S) = Cat(S^{op},Set)$, presheaves on some small category $S$. In particular, we know that $S=\Delta$ can be used to recover the homotopy theory of CW-complexes. The question was to characterise those $S$ such that $(Pre(S),W')$, where $W'$ was inherited from a basic localizer (consult Cisinski's or Maltsiniotis' work for details), can be used to model the same homotopy types as $Cat$. Such categories $S$ were called [weak/strict] test categories.

D. Cisinski, _Les préfaisceaux comme modèles des types d’homotopie_, Astérisque 308 (2006)

and

G. Maltsiniotis, _La théorie de l’homotopie de Grothendieck_, Astérisque, 301 (2005)

are central resources in this area.