I want a bit clarity. In Hartshornes book he states
if we have an exact sequence $\cdots\rightarrow \mathcal{F}^{n-1} \rightarrow^{f^{n-1}} \mathcal{F}^{n} \rightarrow^{f^n} \mathcal{F}^{n+1} \rightarrow^{f^{n+1}} \cdots$ is an exact sequence of sheaves if Ker $f^{n} =$ img$f^{n-1}$ as sheafs sheaves.
Now, Hartshorne is not clear if he is talking about an equality or an isomorphism. It is my belief these are fundamentally different. Furthermore, when he is stating the Kernal sheaf for $f^n$ is he talking explcity about, lets say an open set $U$, Ker $f^{n}(U)= \text{ker}(f^n(U))$. Or is he talking about the assosiated sheaf i.e maps $s:U\rightarrow \bigcup_{p \in U} (\text{Ker }f^n)_p $.
I fully understand as these to sets are isomorphic. But I need clarification of how these definitions build to the exactness definition. I hope what I am asking is clear to the reader. I am more than willing to clarify.
The kernel as you have defined it is already a sheaf so there is no need for sheafification.
Note that your $Ker f^{n}$ is a subsheaf of $\mathcal{F}^{n}$. Thus, to get a sensible definition of exactness at $\mathcal{F}^{n}$, you have to express the image sheaf of $f^{n-1}$ as a subsheaf of $\mathcal{F}^{n}$, so that $Im f^{n-1} = Ker f^{n}$ will make sense as an equality of subsheaves of $\mathcal{F}^{n}$.
For the image, well if $\phi : \mathcal{F} \to \mathcal{G}$ is a morphism of sheaves on some topological space X, then for any open subset $U$ of $X$, we have the following description of the image sheaf of $\phi$. (This is alluded to in exercise 1.4.(b) page 66 of Hartshorne's book.) [I will write $U \mapsto (Im_{Sh} \phi) (U)$ for the image sheaf of $\phi$.]
$(Im_{Sh} \phi) (U) = \{\tau \in \mathcal{G}(U) \:\lvert\:$there exists some open cover $\{U_{}i\}_{i \in I} $ of $U$ and there exists for every $i \in I$ a section $\sigma_{i} \in \mathcal{F} (U_{i}) $ such that for every i, we have $\tau \vert_{U_{i}} = \phi_{U_{i}} (\sigma_{i}) \}$
Morally this means that a section $\tau \in \mathcal{G}(U)$ is in $(Im_{Sh} \phi) (U)$ if and only if it is locally (i.e. on some open cover $ \{U_{i}\} $ ) the image by $\phi_{U_{i}}$ of some section $\sigma_{i} \in \mathcal{F} (U_{i})$.
I agree very much with you that it is not clear if Hartshorne talks about an isomorphism or an equality.
(Note that exactness of maps of sheaves is imo best understood in terms of stalks.)