$\newcommand{\fs}{\mathscr{F}}\newcommand{\gs}{\mathscr{G}}$Let ${\bf Ring}$ be the category of rings, and ${\bf Sh}(X,{\bf Ring})$ the category of sheaves of rings on $X$.
Let $\phi:\fs\to\gs$ be an epimorphism in ${\bf Sh}(X,{\bf Ring})$. Is it true that $\phi_x:\fs_x\to\gs_x$ is an epimorphism in ${\bf Ring}$ for every $x\in X$ ?
Note that epimorphism in ${\bf Ring}$ is not necessarily surjective.
If instead of ${\bf Ring}$, we use the category of abelian groups, then the statement is a familiar result (true). The main trouble seems to be that, for ${\bf Ring}$, there is no zero object, and we can't define things like kernel and cokernel.
Yes. Taking stalks is a left adjoint functor, and left adjoints preserve epimorphisms. As for detecting epimorphisms, you don't need to use cokernels. (Thinking in such terms is a symptom of working with additive categories too much!) Instead you can use the cokernel pair. That is why being an epimorphism is preserved by left adjoints.