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: Regarding question 2, if I am not mistaken, there is a map

$\mu: F^\sharp \to \bigoplus_l F^{l\sharp}$

which maps the germ $<U,s>$ of a section $s$ of $F$ over $U$ to its components, which are germs of sections of $F^l$. One can define a map

$\nu: \bigoplus_l F^{l\sharp} \to F^\sharp$

using the fact that all but finitely many of the components of an element of a direct sum are $0$. The issue I am worried about is that if you have an infinite collection of open sets, their intersection need not be open. But in this case, this issue does not actually occur, because almost all components of an element of a direct sum are $0$.

Moreover, $\mu$ and $\nu$ are inverses of each other. Please check the details (I am not a sheaf expert either).