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To complement Sándor's answer, let me focus on (3) since that seems to be what you are interested in. Let $F$ be the sheaf of generalized functions or distributions. Consider the localization sequence $$ H^0(X,F)\to H^0(X-x,F)\to H_x^1(X, F)\to H^1(X,F)$$ To ensure vanishing of the 3rd term, you would need to know that any distribution on $X-x$ extends to $X$. But surely this isn't true. (E.g. take $1/|x|$ viewed as a distribution on $X-x=\mathbb{R}-\{0\}$ and try to extend.; see Igor's comment, below) On the plus side, since $F$ is soft, the continuation of the same sequence shows that $H^i_x(X, F)=0$ when $i\ge 2$.