As the question title suggests, what is the role cohomology of coherent sheaves plays for SGA 4.5, étale cohomology? Why are they so important for the construction and establishing properties of étale cohomology? What are some of the main essences, intuitions, and theorems of the cohomology of coherent sheaves that are exploited or get "carried over" from the classical setting, in some sense?



"Role" of cohomology of coherent sheaves in SGA 4.5, étale cohomology
You ask two different questions, I believe.


1) When Grothendieck invented étale cohomology, cohomology of coherent sheaves was already well known, thanks to Oka/Cartan's theorems A and B, Serre's FAC, Gaga, and the Serre-Grothendieck duality theorems). It was also well known (Weil, already) that the invention of some cohomology theory closer to singular cohomology was one path towards the Weil conjecture. In some sense, cohomology of coherent sheaves was an insufficient prototype.


2) Coherent sheaves naturally furnish examples of étale sheaves and Grothendieck's descent theorems imply that nothing is gained by viewing them in the étale framework. This implies in particular that étale line bundles or étale vector bundles are the same objects than line bundles or vector bundles. This gives at least some étale sheaves, such as $(\mathbf G\_m)\_X$ or $\mathrm{GL}(n)\_X$, whose cohomology can be computed; for example:
$$ H^1\_{\rm ét}(X,\mathbf G\_m)=H^1\_{\rm Zar}(X,\mathbf G\_m)=\mathop{\rm Pic}(X).$$


For $n$ invertible on $X$, the Kummer exact sequence 
$$ 1\to (\mu\_n)\_X \to (\mathbf G\_m)\_X\xrightarrow{x\mapsto x^n}(\mathbf G\_m)\_X\to 1 $$
gives rise to a long exact sequence which is the starting path to the computation of étale cohomology groups with coefficients in $\mu\_n$.


When $X$ is an $\mathbf F\_p$-scheme, the Artin-Schreier exact sequence
$$ 0 \to (\mathbf Z/p\mathbf Z)\_X \to (\mathbf G\_a)\_X \xrightarrow{x\mapsto x^p-x} (\mathbf G\_a)\_X\to 0 $$
allows to compute the étale cohomology groups of $X$ with coefficients in $\mathbf Z/p\mathbf Z$. One sees in particular that they vanish beyond the dimension of $X$, hence are insufficient for the Weil conjectures.


