Does there exist a group scheme of finite type over a field $k$ , which is neither affine, nor an abelian variety?
For me the most natural examples to think about are those coming from linear algebraic groups in prime characteristic, in the form of Frobenius kernels. These are extensively treated in Jantzen's book Representations of Algebraic Groups, with foundations laid in Part I. Concretely, you can start with the restricted universal enveloping algebra of the Lie algebra of such a group and associate to it naturally a finite group scheme. This is only the "first" Frobenius kernel.
Examples of this kind are "affine" but not linear algebraic groups.
Perhaps more in the spirit of the question, natural examples occur in connection with Chevalley's classical structure theorem: for a modern treatment see B. Conrad, "A modern proof of Chevalley's theorem on algebraic groups", J. Ramanujan Math. Soc. 17 (2002).