I'm trying to take familiarity with homotopy theory and I have the following questions. Let $\mathcal{C}$ be a small category, and let $F\: : \:\mathcal{C}\to \mathcal{M}$ that take values in a (simplicial) model category.
For any simplicial presheaves $K\: : \: \mathcal{C}^{op}\to sSet$, I denote with
$$
K\otimes_{\mathcal{C}} F
$$
the functor tensor product. Let * be the constant simplicial presheaf, it is easy to note that
$$
*\otimes_{\mathcal{C}} F=colim F
$$
Now choose a cofibrant replacement P of * in the projective model structure of simplicial presheaves. If we assume that $F$ is pointwise cofibrant, then the homotopy colimit is given by
$$
P\otimes_{\mathcal{C}} F
$$
A good candidate for the cofibrant replacement for $*$ is the nerve functor $N(-/ \mathcal{C})$. Another strategy to compute the homotopy colimit is given by the bar construction $B(K, \mathcal{C}, F)$ (see for example www.math.harvard.edu/~eriehl/hocolimits.pdf or the book *categorical homotopy theory*). In particular it is possible to show that there is a natural isomorphism
$$
B(*, \mathcal{C}, F)\to N(-/ \mathcal{C})\otimes_{\mathcal{C}} F
$$
Thus in some sense the bar construction contains the data of a cofibrant replacement for the simplicial presheaf $*$.
Here are my questions:

1) Is it possible to generalize the above facts for a general simplicial presheaf $K\: : \: \mathcal{C}^{op}\to sSet$, i.e there exists a cofibrant replacament QK of K and a natural isomorphism $$ B(K, \mathcal{C}, F)\to QK\otimes_{\mathcal{C}} F? $$

2) If not, it is possible to find at least a weak equivalence?

Another formulation of this question is as follows: since $K\otimes_{\mathcal{C}} F$ may be intepretated as a weighted colimit, does the bar construction compute the *homotopy* weighted colimits?

3)(i don'think make sense) If $\mathcal{M}$ is not a simplicial category, it is possible to find a relation between the homotopy weighted colimit and the Bar construction?