I believe I have the answer in the setting of sheaves of sets.
Let us first do this for presheaves, $Set^{C^{op}}$. This category is Cartesian-closed. This can be seen by setting $Y^X(U):=Hom(X \times U, Y)$, where I have identified $U$ with the representable presheaf $Hom(\cdot,U)$. This is easy to verify. It suffices to show that $Y^X$ agrees with the presheaf $U \mapsto Hom(X|_U,Y|_U)$.
As Peter pointed out, we have the functor $l_U:Set^{C^{op}} \to Set^{C^{op}}/U$ which sends a presheaf $X \mapsto \left(X \times U \to U\right)$, which is right adjoint to the corresponding forgetful functor $Set^{C^{op}}/U \to Set^{C^{op}}$. Here, $X|_U:=l_U(X)$. To explain the notation, note that we have an equivalence of categories $Set^{C^{op}}/U \cong Set^{\left(C/U\right)^{op}}$, so we can think of $l_U$ as restricting $X$ to a presheaf over the slice category $C/U$. Now, given $X$ and $Y$ presheaves on $C$, $Hom(l_U(X),l_U(Y))\cong Hom(X \times U, Y)$ since $l_U$ is a right adjoint to the forgetful functor and the forgetful functor applied to $l_U(X)$ is simply $X \times U$. Hence, we see that $U \mapsto Hom(X|_U,Y|_U)$ agrees with the functor $U \mapsto Hom(X \times U, Y)$.
I claim the same works for a Grothendieck topos:
For this, it suffices to prove that the functor $U \mapsto Hom(X \times U, Y)$ is a sheaf whenever $Y$ is. Let $\left(s_i:U_i \to U\right)_i$ be a cover of the object $U$. Note that $\left(s_i \times id:U_i\times X \to U\times X\right)_i$ is a cover of $U\times X$.
So $\varprojlim \left( \prod \limits_i Y^X(U_i) \rightrightarrows \prod \limits_{i,j} Y^X(U_i\times_U U_j)\right)\cong \varprojlim \left( \prod \limits_i Hom(U_i\times X,Y) \rightrightarrows \prod \limits_{i,j} Hom(U_i\times_U U_j \times X,Y)\right)$
and this is in turn:
$\varprojlim \left( \prod \limits_i Hom(U_i\times X,Y) \rightrightarrows \prod \limits_{i,j} Hom(\left(U_i\times X\right)\times_{U\times X} \left(U_j \times X\right),Y)\right) \cong Hom(U \times X, Y)$
since $Y$ is a sheaf.
Q1: A very simple example is given in Grothendieck's Tohoku paper "Sur quelques points d'algebre homologiquie", sec. 3.8. Edit: The space is the plane, and the sheaf is constructed by using a union of two irreducible curves intersecting at two points.
Q2: Cech cohomology and derived functor cohomology coincide on a Hausdorff paracompact space (the proof is given in Godement's "Topologie algébrique et théorie des faisceaux"). I don't know of an example on a non paracompact space where they differ.
Best Answer
Artin, M. Grothendieck topologies. (English) Zbl 0208.48701 Cambridge, Mass.: Harvard University. 133 p. (1962). (pdf copy)
These notes seem to fit your description precisely. They are concise, start from first principles, assuming basically only knowledge of Grothendieck's Tôhoku paper. The focus is specifically on how to define cohomology in a topos, as opposed to many references on topos theory aimed more in the direction of algebraic stacks, logic, motivic homotopy...
(This text was recommended in a now deleted answer by another user. In my opinion Artin's notes are quite nice and I thought the recommendation was worth preserving.)