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 $F^+$.