Classification of surjective group homomorphisms

The key fact you need is that if $f: G \rightarrow H$ is a surjective group homomorphism, then it induces a group isomorphism $\tilde{f}: G/ker(f) \rightarrow H$. This gives you that $G_1$ and $G_2$ are isomorphic and also gives you a yes for question 2. and a relation for question 3.

Edit: For question 1, I claim that $h=\tilde{f}_1\circ \tilde{f}_2^{-1}$. The equation $h\circ f_1=f_2$ is then equivalent to $\tilde{f}_1^{-1}\circ f_1=\tilde{f}_2^{-1}\circ f_2$ which holds because both sides are the canonical map from $G$ to $G/ker(f_1)$.