**Regarding Question 1**, the only time I can think of when a Thomason fibrant replacement can be taken to be a 1-categorical localization is when $C$ has the homotopy type of the classifying space of a discrete groupoid $G$, i.e. $|C|$ is _aspherical_.[1] That is:

 - For any category $C$, we have $\Pi_1(|C|) \simeq C[C^{-1}]$, where $\Pi_1$ denotes the fundamental groupoid. (Proof: the van Kampen theorem allows one to write down a presentation of $\Pi_1(|C|)$ from the 2-skeleton of the nerve of $C$, and it is equally a presentation of $\Pi_1(C)$.).

 - So under the assumption that $|C| \simeq BG$, we have $G = C[C^{-1}]$, and therefore $|C| \to |C[C^{-1}]|$ is a weak homotopy equivalence.

 - Moreover, the nerve of a groupoid is a Kan complex (and conversely!), so under the assumption that $|C|$ is aspherical, we have that $C \to C[C^{-1}]$ i a Thomason fibrant replacement.

In particular, **regarding Question 2**, a sufficient condition for $|C| \to |C[S^{-1}]|$ to be a weak homotopy equivalence is for $|C|$ to be aspherical.

This may sound quite restrictive, but in practice it turns out that **most of the categories one actually _likes_ to work with _qua_ categories are aspherical**.

 - For one thing, most of the categories one really cares about [are contractible](https://www.matem.unam.mx/~omar/notes/contractible.html) -- a category is contractible if it has a terminal object, if it has binary products, if it is filtered, if it is homotopy sifted, if it admits an adjunction to such a category, etc.

But more generally, there are some natural categorical conditions which imply that a category is aspherical without necessarily being contractible:

 - For instance, if $C$ has pullbacks, then by a [theorem of Pare](https://doi.org/10.4153/CJM-1990-038-6), $C$ has all finite simply-connected limits. Then arguing using the $Ex^\infty$ functor, one sees that $|C|$ is aspherical.

 - For another example, [Dwyer and Kan showed](https://doi.org/10.1016/0022-4049(80)90113-9) (Prop 7.3 -- this paper follows up on the one linked in the question) that if $C$ has a left calculus of fractions, then $|C|$ is aspherical.[2] More generally, they showed that if $C$ has a left calculus of fractions as well as a left calculus of fractions with respect to a class of morphisms $S$, then then the $\infty$-categorical localization $L_S C$ (=simplicial localization) agrees with the 1-categorical localization $C[S^{-1}]$.

 - There are probably other interesting conditions in this vein which I don't know -- I'd love to hear of more!

Of course, the duals of all of the above statements also hold.

[1] There's a possible confusion here: I'm using "aspherical" in the sense of [aspherical space](https://en.wikipedia.org/wiki/Aspherical_space), i.e. a 1-truncated space. But since we're talking about modeling spaces with categories, it's worth bearing in mind that in this context, the Grothendieck school (I'm thinking of Grothendieck, Maltsiniotis, Cisinski, Ara,...) uses the term "aspherical category" to mean something different (namely, "category with a weakly contractible classifying space" -- possibly after performing an appropriate localization if working with respect to a different [fundamental localizer](https://ncatlab.org/nlab/show/basic+localizer)).

[2] I'm not sure if this is related to the fact, discussed by Meier and Ozornova in the article linked in the question, that $C$ has a left calculus of fractions if and only if $Ex NC$ is a Kan complex.