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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 (nLab). 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 $Cat$ (or $(C,W)$). Cat$. Such categories$S$were called [weak/strict] test categories. D. Cisinski, Les préfaisceaux comme modèles des types d’homotopie, Astérisque 308 (2006) and G. Maltsiniotis, La théorie de l’homotopie de Grothendieck, Astérisque, 301 (2005) are central resources in this area. 1 This is just to answer your first question. The second one I don't know about. The homotopy theory of categories is not quite as you envisage it. Really Grothendieck is thinking of the Thomason model structure on$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$is a class of functors which acted as weak equivalences. This he called a basic localizer (nLab). Grothendieck conjectured, and Cisinski proved, that the class of weak equivalences in the Thomason model structure was the smallest basic localizer. From there Grothendieck moved to considering pairs$(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$(or$(C,W)$). Such categories$S\$ were called [weak/strict] test categories.