For an abelian category $A$, the following are equivalent:

1. Every short exact sequence splits.
2. Every object is projective.
3. Every object is injective.
4. Every additive functor from $A$ to an abelian category is exact.

(To show 4 $\Rightarrow$ 2, use the Hom functor.)

If $A$ has these properties, any full pseudo-abelian subcategory of $A$ is a Serre subcategory (in particular, abelian) with the same properties.

$A$ is semi-simple provided every object of $A$ is Noetherian (or, equivalently, Artinian), or if it is a category of modules over an appropriate additive category. A fun counterexample is the category of infinite-dimensional vector spaces over a field, localised by the Serre subcategory of finite-dimensional vector spaces. I am not sure that it is Grothendieck (does it have infinite direct sums?) Another example, still in the spirit of Leonid's answer, is the category of finitely generated left ideals of a von Neumann regular ring. (Such ideals are generated by an idempotent, hence direct summands, hence their quotients are also finitely generated, etc.)