Let $\mathrm{Ch}_{\geq 0}(R)$ be the category of chain complexes of $R$-modules concentrated in nonnegative degrees, equipped with the projective model structure. By a general theorem about model categories (Chapter 4 in Hovey), the homotopy category $\operatorname{Ho} \mathrm{Ch}_{\geq 0}(R)$ is a right module over $\operatorname{Ho} \mathrm{sSet}$, the homotopy category of simplicial sets (with the Quillen model structure), that is there exists a functor $$ \operatorname{Ho} \mathrm{Ch}_{\geq 0}(R) \times \operatorname{Ho} \mathrm{sSet} \longrightarrow \operatorname{Ho} \mathrm{Ch}_{\geq 0}(R), $$ often written as $(K,X) \longmapsto K \otimes^\mathbb{L} X$, which satisfies several axioms.
I believe that an explicit description of $K \otimes^\mathbb{L} X$ is $$ K \otimes^\mathbb{L} X \stackrel{?}{=} K \otimes_R |R\langle X \rangle|, $$ that is: Turn the simplicial set $X$ into a simplicial $R$-module by applying the free $R$-module functor levelwise, then realize this simplicial module as a chain complex by the Dold–Kan correspondence $|\cdot| : \mathrm{sMod}(R) \to \mathrm{Ch}_{\geq0}(R)$, and finally tensor this complex with $K$ by using the ordinary (non-derived) tensor product of chain complexes.
Question. Is this description correct? Where could I read up on details? I'm slightly puzzled because this observation doesn't seem to appear in the usual references about model categories even though I feel that this ought to be a basic fact.
There's some indication that the description is correct: Quasi-isomorphisms $K \to K'$ induce quasi-isomorphisms $K \otimes^\mathbb{L} X \to K' \otimes^\mathbb{L} X$ (since $|R\langle X \rangle|$ is a complex consisting of projective, thus flat, modules) and $K \otimes^\mathbb{L} (X \times X')$ is quasi-isomorphic to $(K \otimes^\mathbb{L} X) \otimes^\mathbb{L} X'$ by the Eilenberg–Zilber map.
Bonus question. Is $\mathrm{Ch}_{\geq0}(R)$ a right module over $\mathrm{sSet}$? If so, is the action given by the same formula?
Hovey denies this (page 114, "so far as the author knows there is no monoidal Quillen functor $\mathrm{sSet} \to \mathrm{Ch}(R)$") and at least the stated description shouldn't work, since the Dold–Kan functor preserves the tensor product only up to homotopy equivalence, not up to isomorphism. However, I believe that the nLab (Remark 2) and these nice notes (Exercise 1.2.2) do claim this.