Is epimorphism preserved under taking sections?

No. A counterexample (specialised from Gunning/Rossi, Analytic Functions of Several Complex Variables, Chapter VI):

Let $D$ an open interval in $\mathbb{R}$, and $\mathscr{F}$ the sheaf of germs of constant (complex valued) functions ($\mathscr{F}_z \cong \mathbb{C}$ for all $z \in D$). Let $\mathscr{K}$ be the subsheaf of $\mathscr{F}$ with stalk $0$ in two points $a < b$, and stalk $\mathbb{C}$ everywhere else. Let $\mathscr{G}$ be the quotient sheaf $\mathscr{F}/\mathscr{K}$. $\mathscr{G}$ has stalk $0$ above all points except $a$ and $b$, where it has stalk $\mathbb{C}$.

We have a short exact sequence

$$0 \to \mathscr{K} \to \mathscr{F} \to \mathscr{G} \to 0$$

of sheaves, but $\Gamma(U,\,\mathscr{F}) \to \Gamma(U,\,\mathscr{G})$ is not surjective for any connected $U$ containing both $a$ and $b$, since a section of $\mathscr{G}$ can take different values in $a$ and $b$, but a section of $\mathscr{F}$ must be constant.

If you want to avoid $\{0\}$-rings, consider

$$0 \to \mathscr{K} \oplus 0 \to \mathscr{F} \oplus \mathscr{F} \to \mathscr{G} \oplus \mathscr{F} \to 0.$$

For sheaves above paracompact Hausdorff spaces $D$, a short exact sequence

$$0 \to \mathscr{R} \to \mathscr{S} \to \mathscr{T} \to 0$$

induces a short exact sequence

$$0 \to \Gamma(D,\, \mathscr{R}) \to \Gamma(D,\,\mathscr{S}) \to \Gamma(D,\,\mathscr{T}) \to 0$$

for example if $\mathscr{R}$ is a soft sheaf (any section over a closed subset $K \subset D$ can be extended to a section over all of $D$).