Since Čech cohomology is mentioned, I presume that $C$ is the category of open subsets of a topological space.

In this case, the answer to both questions is yes. It follows from a more general result:

If $G$ is a sheaf and $f\colon P\to Q$ is a local isomorphism of presheaves (i.e., a morphism that induces an isomorphism on all stalks), then the induced map $$\def\Hom{\mathop{\rm Hom}} \Hom(Q,G)→\Hom(P,G)$$ is an isomorphism.

This abstract result applies to the two cases under consideration because the maps $F^+→F^\sharp$ and $F\to (F^\sharp)^+=F^\sharp$ are local isomorphisms.

Indeed, even more generally, the natural map $F\to F^+$ is a local isomorphism for any presheaf $F$. This follows immediately from the fact that the stalk functor is cocontinuous, in particular, it preserves the colimit used to define $F^+$.

Since Čech cohomology is mentioned, I presume that $C$ is the category of open subsets of a topological space. More generally, we can assume $C$ to be an arbitrary site.

In this case, the answer to both questions is yes. It follows from a more general result:

If $G$ is a sheaf and $f\colon P\to Q$ is a local isomorphism of presheaves (i.e., a morphism that becomes an isomorphism after passing to associated sheaves; in the case of sites with enough points, such as topological spaces, it can be described as a morphism that induces an isomorphism on all stalks), then the induced map $$\def\Hom{\mathop{\rm Hom}} \Hom(Q,G)→\Hom(P,G)$$ is an isomorphism.

This abstract result applies to the two cases under consideration because the maps $F^+→F^\sharp$ and $F\to (F^\sharp)^+=F^\sharp$ are local isomorphisms.

Indeed, even more generally, the natural map $F\to F^+$ is a local isomorphism for any presheaf $F$. Indeed, the associated sheaf functor can be computed as $F↦F^{++}$, so if we apply it to the morphism $F→F^+$, we get the identity map $F^{++}→F^{+++}=F^{++}$.

In the case of sites with enough points, this also follows immediately from the fact that the stalk functor is cocontinuous, in particular, it preserves the colimit used to define $F^+$.