The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories?

In fact, applying the simplicial construction you describe to a category $\mathcal{C}$ gives the homotopy coherent thickening $\mathfrak{C} N\mathcal{C}$ (where $N$ is the nerve and $\mathfrak{C}$ is the left adjoint to the homotopy coherent nerve, as in HTT). This is described in Emily Riehl's paper "On the structure of simplicial categories associated to quasicategories" (see Theorem 6.7).

As a result, since $\mathfrak{C}$ of any simplicial set is cofibrant, $\mathfrak{C}N \mathcal{C}$ certainly is (in the Bergner model structure). The fact that $\mathfrak{C } N \mathcal{C} \to \mathcal{C}$ is a weak equivalence follows because $N \mathcal{C}$ is a fibrant object in the Bergner model structure, so to say that the above map is a weak equivalence is to say that $N \mathcal{C} \to N \mathcal{C}$ is one. Here I have used two facts: that the nerve of an ordinary category is the same as its homotopy coherent nerve and the (much harder) fact that $\mathfrak{C}$ and the homotopy coherent nerve determine a Quillen equivalence between the Joyal and Bergner model structures (or at least HTT 2.2.0.1 suffices).