Let $X$ be a smooth manifold. Let $F$ be a sheaf of $\mathbb{R}$-vector spaces on $X$. I have three closely related questions.

1. Under what sufficient conditions on $F$ for any compact subset $K\subset X$ one has $$R^i\Gamma_K(X,F)=0\mbox{ for all } i>0,\,\,\,\,(1)$$ where $\Gamma_K$ is the functor of global sections supported on $K$, and $R^i\Gamma_K$ are its right derived functors?

2. Under what sufficient conditions on $F$ one has $$R^i\Gamma_{\{x\}}(X,F)=0\mbox{ for all } i>0,\,\,\,(2)$$ for any point $x\in X$.

3. Let $F$ be the sheaf of generalized functions on $X$. Is it true that either (1) or (2) are satisfied for this particular $F$?

If the answer to question (3) is negative, then probably I do not need to answer questions (1), (2) since I will not be able to use that. But if it is positive, then the answer to (1) or (2) should contain the answer to (3) as a special case.

Let $X$ be a smooth manifold. Let $F$ be a sheaf of $\mathbb{R}$-vector spaces on $X$. I have three closely related questions.

1. Under what sufficient conditions on $F$ for any compact subset $K\subset X$ one has $$R^i\Gamma_K(X,F)=0\mbox{ for all } i>0,\,\,\,\,(1)$$ where $\Gamma_K$ is the functor of global sections supported on $K$, and $R^i\Gamma_K$ are its right derived functors?

2. Under what sufficient conditions on $F$ one has $$R^i\Gamma_{\{x\}}(X,F)=0\mbox{ for all } i>0,\,\,\,(2)$$ for any point $x\in X$.

3. Let $F$ be the sheaf of generalized functions on $X$. Is it true that either (1) or (2) are satisfied for this particular $F$?