If I is cofibrant and all objects are fibrant,
then Map(I,−) is automatically derived.
Thus, precomposing with the cofibrant replacement functor Q
suffices to derive both |F| and F.
We get L|F|=|QF| and |LF|=|QF|,
so indeed there is a weak equivalence between L|F| and |LF|.
I'm not very knowledgeable on model categories, so I don't know how this relates to your precise claim, however the claim the nLab makes (homotopy product = product in the homotopy category) is quite easy to prove (the reason I don't know how this relates to your claim is that you are considering $\mathbb L(S\times -)$, whereas the homotopy product is the right derived functor of the product functor $\mathbf{sSet}^2 \to \mathbf{sSet}$).
Let $C$ be a model category. Unless indicated otherwise, $\times$ denotes the usual product in $C$; $[-,-]$ denotes the hom-set in the homotopy category.
Let $X,Y$ be fibrant, then clearly the image of $X\times Y$ is the product of $X$ and $Y$ in the homotopy category : if $Z$ is any object, then one may cofibrant replace it and so assume that $Z$ is cofibrant, so $[Z, X\times Y]$ is isomorphic to homotopy classes of maps $Z\to X\times Y$ and therefore to pairs of homotopy classes of maps $Z\to X, Z\to Y$ (here we use the fact that we can choose a cylinder object for $Z$ uniformly, and any homotopy can be realized via this fixed cylinder object), and therefore to $[Z,X]\times [Z,Y]$.
Therefore, if $X,Y$ are any objects, then you fibrant replace them with $QX,QY$ and get that their homotopy product is $QX\times QY$ which also happens to be the product of $QX$ and $QY$ in the homotopy category, which is therefore the product of $X$ and $Y$ in the homotopy category (because $X\simeq QX, Y\simeq QY$ in the homotopy category obviously)
So $X\times_h Y = X\times_{\mathrm{Ho}(C)} Y$
Best Answer
As you say, you have choices. Normally for homotopy colimits you want the projective model structure (if that is available). Sometimes the Reedy model structure works too. There is also the injective model structure, which is used for homotopy limits.
To give more details, if $\mathsf{C}$ is cofibrantly-generated, the projective model structure on $\mathsf{C}^\mathsf{I}$ has objectwise weak equivalences, objectwise fibrations, and cofibrations whatever they have to be. Dually for the injective model structure. The Reedy model structure takes a bit longer to describe.
You may also be interested in techniques to transfer model structures along adjunctions.
FYI, I like Dugger's (unfinished) notes on homotopy colimits as a reference. See part 3 for model structures on diagram categories.