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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.

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

Regarding 1, 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.

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 $X$ (maybe that $X$ is connected?)? We need to fix this post.

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.

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 1 (itself edited): Regarding question 2, if I am not mistaken, there is a map

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

which, given an open set $U$, gives a map

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

defined as follows. Consider an element $s \in F^\sharp(U)$, which can be thought of as sending a point $x \in U$ to $s(x) \in F_x$ such that, given a point $x_0 \in U$, there is an open set $V \subseteq U$ containing $x_0$, and an element $t \in F(V)$, such that for any $x \in V$, $s(x)$ is equal to the germ $t_x$ of $t$ at $x$.

But $t$ can be written as $t = (t^l)$, where each $t^l \in F^l(V)$, and all but finitely many of the $t^l$ are $0$. One can map each $t^l$ to an element $s^l \in F^{l\sharp}(V)$, defined by requiring that $s^l(x) = t^l_x$. Just like in my answer for question 1, using the universal property of direct sums (which are coproducts in the category of abelian groups), one may map $t$ to $(s^l)$, which defines a map

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

The map $\mu$ induces an isomorphism at the level of stalks. Indeed, for any $x \in X$, $F^\sharp_x \simeq F_x$, and $F^{l\sharp}_x \simeq F^l_x$, and $\mu$ is easily seen to induce an isomorphism from $F^\sharp_x$ onto $(\bigoplus_l F^{l\sharp})_x$, itself isomorphic to $\bigoplus_l F^l_x$. So $\mu$ is an isomorphism of sheaves.

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 $X$ (maybe that $X$ is connected?)? We need to fix this post.