Let $\mathcal{C}$ and $\mathcal{D}$ be abelian (resp. additive) categories. Does there exist a smallest abelian (resp. additive) category $\mathcal{C}+\mathcal{D}$, s.t. $\mathcal{C}$ and $\mathcal{D}$ embed "in a reasonable way" into $\mathcal{C}+\mathcal{D}$?
It should be the 2-categorical union (citation needed) in the 2-category of abelian (resp. additive) categories.
Yes, it is just the product $\mathcal{C} \times \mathcal{D}$. It is a categorification of the usual construction of the coproduct of abelian groups. The inclusions are $\mathcal{C} \to \mathcal{C} \times \mathcal{D}$, $X \mapsto (X,0)$ and $\mathcal{D} \to \mathcal{C} \times \mathcal{D}$, $Y \mapsto (0,Y)$. If $F : \mathcal{C} \to\mathcal{E}$, $G : \mathcal{D} \to \mathcal{E}$ are two additive functors, we get an additive extension $(F;G) : \mathcal{C} \times \mathcal{D} \to \mathcal{E}$, $(X,Y) \mapsto F(X) \oplus G(Y)$. If $F$ and $G$ are exact, then $(F;G)$ is exact as well. This construction provides a bicategorical coproduct (not a $2$-categorical coproduct) of $\mathcal{C},\mathcal{D}$ in the $2$-category of additive categories with additive functors as well as the $2$-category of abelian category with exact functors.