Are there (infinite) non-isomorphic groups $G, H$ such that there are surjective group homomorphisms $f: G\to H$ and $g: H\to G$? 



Non-isomorphic groups such that there are epis from one to another
Yes, an example is $\mathbb{Z}/2 \times \mathbb{Z}^{\mathbb{N}}$ and $\mathbb{Z}^{\mathbb{N}}$. This even works in the category of rings.


