Definition of a Sheaf, making sense of I have started to read an english translation of Serre's FAC. Immediately a sheaf is defined. The more categorical definition given on wikipedia actually makes more sense to me, but I would like to understand this one as it is written since I will be working through this paper.
I give the definition, then pose some questions.
Definition:

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
  $\mathscr{F}+\mathscr{F}$.
Having stated the above definitions, we impose two axioms on the data
  (a) and (b):
(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
  $\mathscr{F}$.

Questions:

1) At the very beginning of this definition, what are we taking
  $\mathscr{F}$ to be? It seems we are referring to it indirectly as the
  category of abelian groups. In this definition is $\mathscr{F}$ just
  some unspecified class of abelian groups? They refer to $\mathscr{F}$ as a set in (b), but I don't believe the category of abelian groups is small. What should I be taking $\mathscr{F}$ as?
2) After this definition, which thing(s) exactly is (are) the sheaf?
  Is it the pair $(f,\tau)$ where $f$ is the function from (a) and
  $\tau$ the topology from (b)? The function from (a) seems to be
  playing the role of the functor from the definition on wikipedia,
  except that the functor pairs open sets of $X$ to objects, not points of the
  space $X$.
3) In part (b), a priori of the rest of the definition, is it just
  stating that any topology can be on $\mathscr{F}$? Why do they say
  'the sum of the sets $\mathscr{F}_x$'.
4) In I and II, how am I to make sense of $-f$ and $f+g$ if $f,g$ are
  abelian groups? does this have to do with the topology we put on the
  collection of sets?
5) Is this definition actually equivalent to the one on wikipedia? Of
  course this is just for abelian groups, and the one on wikipedia
  allows the target category to be Sets, Rings, etc.. But in the case of
  the target category being abelian groups, are they equivalent?

EDIT:


*

*Perhaps question 3 actually answers question 1. Are they defining the 'set' $\mathscr{F}$ to be the sum of all the images of the function? In what set theoretic way are we summing them? The disjoint union seems to be plausible?

*Also, -f and f+g do make sense, I just realized that $f,g$ are elements of the group $\mathscr{F}_x$.

*In light of these edits, can I take $$\mathscr{F} = \bigcup_{x \in X} (\{x\} \times \mathscr{F}_x) \hspace{2mm} ?$$ then the mapping $\pi$ mentioned in the definition would really be mapping from $(x,f) \mapsto x$? Although maybe trivial at this level, this seems important since in the case of the constant sheaf the same group element would be going to each $x \in X$ but in this disjoint union that would still be well defined.

*The definition of the map $\pi$ seems very 'not universal' since it depends on the elements of the groups? 

*The definition of $\mathscr{F} + \mathscr{F}$ seems very strange, it is just forming all tuples of elements that came from the same group?
 A: 1) In (b), "sum" means disjoint union. So $\mathscr{F}$ is the union of the groups $\mathscr{F}_x$, which are the "stalks" of the sheaf.
2) In this definition, the sheaf is the space $\mathscr{F}$, with the appropriate topology. It is also common to say that the sheaf "is" the functor sending an open subset $U \subset X$ to the set $\mathscr{F}(U)$ of continuous sections $U \to \pi^{-1}(U)$, which in fact has the structure of an abelian group by axiom (II).
3) Any topology on $\mathscr{F} = \bigsqcup_x \mathscr{F}_x$ satisfying axioms (I) and (II) is a valid topology defining a sheaf.
4) You seem to have figured this one out already.
5) Yes. See 2) for what the functor should be. 
You should keep in mind the simplest example of a sheaf, given directly below the definition, which is $\mathscr{F} = X \times G$ for an abelian group $G$, where $G$ is given the discrete topology. So here $\mathscr{F}_x = G$ for every $x$. Translating to the language of a functor on open sets, this is the sheaf of locally constant $G$-valued functions. 
