Let $A/k$ be a simple abelian variety.
Does there exist a non-simple abelian variety $B/k$ and a finite homomorphism $f:B\to A$ over $k$?
I don't need $f:B\to A$ to be etale.
No. If $End_A$ and $End_B$ are the endomorphism rings then an isogeny $B\to A$ will give an isomorphism $End_A\otimes\mathbb Q\to End_B\otimes\mathbb Q$. But the first of these algebras is a division algebra and the second is not.