The answer to 1 was too long as a comment, so I will write it here.

Regarding 1, I think the answer is 'yes', meaning that one can define the notion of a direct sum of presheaves of abelian groups, but please check the details. 

Suppose we have presheaves $F^l$ of abelian groups on a topological space $X$. We want to define their direct sum

$F = \bigoplus_l F^l$

First, we define, for $U$ an open subset of $X$,

$F(U) = \bigoplus_l F^l(U)$ .

Moreover, if $V \subseteq U$ is an open subset of $X$ contained in $U$, we denote the restriction homomorphisms of $F^l$ by $h^l_{VU}$, so that

$h^l_{VU}: F^l(U) \to F^l(V)$ .

We now define the restriction homomorphism $h_{VU}: F(U) \to F(V)$ as follows. Let $i_k(W): F^k(W) \to \bigoplus_l F^l(W)$ be the natural homomorphisms, where $W$ is an arbitrary open subset of $X$.

Consider the homomorphisms $g^l_{VU} = i_l(V) \circ h^l_{VU}: F^l(U) \to F(V)$. By the universal property of coproducts, satisfied by direct sums of abelian groups, there is a unique homomorphism $h_{VU}: F(U) \to F(V)$ such that $g^l_{VU} = h_{VU} \circ i_l(U)$ for any $l$.

Thus we have defined restriction homomorphisms $h_{VU}$, and it remains to check that they satisfy the required properties.

Edit: I am not sure about my answer for question 2, so I deleted it. What I now know is that there exist families of sheaves whose direct sum, as a presheaf, does not satisfy the gluing axiom, and thus is not a sheaf. @Max, does the statement "sheafification is a left-adjoint functor with respect to the inclusion of the category of sheaves in the category of presheaves", assumes some conditions on the topological space $X$? We need to fix this post.