Serres gives the following definition of a sheaf in the paper FAC:
Let $X$ be a topological space. A sheaf of abelian groups on $X$ (or simply a sheaf ) consists of:
(a) A function $x \to \mathscr{F}_x$, giving for all $x \in X$ an abelian group $\mathscr{F}_x$,
(b) A topology on the set $\mathscr{F}$, the sum of the sets $\mathscr{F}_x$.
If $f$ is an element of $\mathscr{F}_x$, we put $\pi(f) = x$; we call the mapping of $\pi$ the projection of $\mathscr{F}$ onto $X$; the family in $\mathscr{F} \times \mathscr{F}$ consisting of pairs $(f,g)$ such that $\pi(f) = \pi(g)$ is denoted by $\mathscr{F}+\mathscr{F}$.
(I) For all $f \in \mathscr{F}$ there exist open neighborhoods $V$ of $f$ and $U$ of $\pi(f)$ such that the restriction of $\pi$ to $V$ is a homeomorphism of $V$ and $U$.(In other words, is a local homeomorphism).
(II) The mapping $f \mapsto -f$ is a continuous mapping from $\mathscr{F}$ to $\mathscr{F}$, and the mapping $(f, g) \mapsto f + g$ is a continuous mapping from $\mathscr{F}+\mathscr{F}$ to $\mathscr{F}$.
If $U$ is an open subset of $X$ then a map $s: U \to \mathscr{F}$ is called a section over $U$ if $s$ is continuous and $\pi \circ s =$ id$_U$.
It is asserted that the set of all sections over a fixed subset $U$ form an abelian group with the operation of pointwise addition. I would like to like to verify this. I see that if there is at least one section $s$ on $U$, the abelian group structure will follow from (2). But how do I know there is at least one section, say $s$, for each $U$?
I tried to prove that for a given $U$, the map that takes $x \in U$ to the identity element of the corresponding stalk, $(x,e)$ is continuous, by invoking 1, but was unsuccessful.
Consider the mapping you gave, that is $x\mapsto (x,e_x)$ where $e_x$ is the identity element of $\mathscr{F}_x$.
You want to show that it is continuous, but continuity is a local property so you can look locally; and then invoke (I) :
Let $x\in X$ and let $V$ be a neighbourhood of $(x,e_x)\in \mathscr{F}$ and $U$ a neighbourhood of $x$ such that $\pi: V\to U$ is a homeomorphism. Let $s:U\to V$ be its converse.
Denote $V\times_X V = \{(z,y) \in V, \pi(z) = \pi(y)\}$ (this is the more common notation for $V+V$). By (II), $m:V\times_X V \to V$ defined by $m(z,y) = z-y$ is continuous.
Consider now $\varphi: V\to V\times_X V$, $y\mapsto (y,y)$ which is also continuous.
Finally, let $d: U\to V$ be defined as $m\circ \varphi \circ s$. $d$ is continuous, $\pi\circ d = id_U$ is also clear from the definitions.
Moreover, unravelling the definition yields that $d:U\to V$ is precisely $d(y)= (y,e_y)$: hence $x\mapsto (x,e_x)$ is locally continuous, hence continuous.
So this gives us a global section $X\to \mathscr{F}$ which is a neutral element in the set of sections of $X$, and clearly its restriction to any open set has the same property.
Passing remark: In the functorial definition, a sheaf of groups is a functor $O(X)^{op}\to \mathbf{Grp}$ satisfying certain "gluing conditions". But it's actually easy to see that it's the same thing as a group object in the category $\mathbf{Sh}(X)$, the category of sheaves of sets on $X$. So with this definition, the continuity of the aforementioned map is automatic, because in the definition of a group object you have a map from the terminal object to the group $G$, but this essentially means a map $X\to G$ "over $X$" (seeing $\mathbf{Sh}(X)$ as $Etale(X)$)