For a closed subset $Z$ of a topological space $X$ and for a sheaf of abelian groups $\mathcal F$ on $X$, let $\Gamma_Z(X,\mathcal F)$ be the subgroup of $\Gamma (X,\mathcal F)$ consisting of all sections whose support is contained in $Z$.
Then is it true that $\Gamma_Z(X,\mathcal F)=\Gamma_Z(U,\mathcal F|_U)$ for every open set $U$ of $X$ with $Z \subseteq U$ ?
In particular, if $x\in X$ is a closed point and $Z:=\{x\}$ , and $\underline {\Gamma_Z} (\mathcal F)$ is the sheaf that sends
$V\to \Gamma_{V\cap Z}(V,\mathcal F|_V)$ , then is it true that $\underline {\Gamma_Z} (\mathcal F)_x=\Gamma_Z (X,\mathcal F)$ ?
Yes, it is true that $\Gamma_Z(X,\mathcal{F})=\Gamma_Z(U,\mathcal{F}|_U)$ for $U\supset Z$. Clearly any element of the left hand side produces an element of the right hand side by restriction to $U$. Then any element $s$ of the right hand side can be uniquely enlarged to a section $s'$ on $X$ supported on $Z$ by gluing together the section $s$ on $U$ with the section $0$ on $X\setminus Z$ - these agree on the overlap $U\cap (X\setminus Z)$ because both vanish there. But $res_U(s')=s$ and $res_U(s)'=s$, so the sets are the same. This easily implies your final claim about the stalk of the sheaf.