Is taking cokernels coproduct-preserving? Let $\mathcal{A}$ be an abelian category, $A\,A',B$ three objects of $\mathcal{A}$ and $s: A\to B$, $t: A' \to B$ morphisms.
Is the cokernel of $(s\amalg t): A\coprod A'\to B$ the coproduct of the cokernels of $s$ and $t$?
In case it's wrong: Is it true if we restrict $s$ and $t$ to be monomorphisms?

Colimits preserve colimits, so colimits do preserve coproducts, and cokernels are colimits. However, this means something different than what you suggest.
Usually, $s \coprod t$ is used to mean the morphism $A \coprod A' \to B \coprod B$; with this meaning, we do have
$$ \text{coker}(s \coprod t) = \text{coker}(s) \coprod \text{coker}(t)$$
If I let $(s,t)$ denote the morphism $A \coprod A' \to B$, then if I've not made an error, what we do have is a pushout diagram
$$ \begin{matrix}
B &\to& \text{coker}(s)
\\\downarrow & & \downarrow
\\\text{coker}(t) &\to& \text{coker}(s,t)
\end{matrix} $$
or equivalently, we have an exact sequence
$$ B \to \text{coker}(s) \oplus \text{coker}(t) \to \text{coker}(s,t) \to 0 $$
