Added even later: Question 1 has been sorted in the affirmative by Dmitri's answer below. Question 2 also has an affirmative answer assuming that the diagram is finite (so say $G$ is a constant presheaf with stalk a finite group). This is Lemma 1.20 in Morel-Voevodsky "$\mathbb{A}^1$-homotopy of schemes". They state it without proof, so I give an elementary one here (leaving the details of a final check out). Do note, it has absolutely zilch to do with filtrant limits.:
The homotopy limit over a finite diagram can be expressed as a limit, over a different, but still finite diagram. Stalks, by definition, commute with finite limits. Hence, all that remains to be done is to check that when we do this commutation the resulting limit (in simplicial sets) is precisely the homotopy limit of the stalk (or that the canonical map between the two simplicial sets obtained is a weak equivalence).
In fact, essentially the same proof should also work for an arbitrary site using pullback to a Boolean localization (but I don’t really understand those, so…).
P.S. In all honesty I have not checked the last `check’. However, at this point I am impatient enough to give Morel-Voevodsky the benefit of the doubt (given that they use the cited result everywhere and it is a landmark paper)Please see Maxime Ramzi's counterexample in the comments to the accepted answer. Either I am missing something, or the claim in Morel-Voevodsky needs refinement.
Do let me know if I said something silly.
The homotopy limit over a finite diagram can be expressed as a limit, over a different, but still finite diagram. Stalks, by definition, commute with finite limits. Hence, all that remains to be done is to check that when we do this commutation the resulting limit (in simplicial sets) is precisely the homotopy limit of the stalk (or that the canonical map between the two simplicial sets obtained is a weak equivalence).
In fact, essentially the same proof should also work for an arbitrary site using pullback to a Boolean localization (but I don’t really understand those, so…).
P.S. In all honesty I have not checked the last `check’. However, at this point I am impatient enough to give Morel-Voevodsky the benefit of the doubt (given that they use the cited result everywhere and it is a landmark paper).