For two elliptic curves $E_1,E_2$, what is the structure of $Hom(E_1,E_2)$ as a left/right module over the endomorphism rings of $E_1,E_2$?

Question

Given two elliptic curves - lets say over $\mathbb{C}$ for simplicity, the group $$Hom(E_1,E_2)$$ is naturally an $(End(E_2),End(E_1))$-bimodule.

What is the structure of this bimodule?

Considered as a module under one of $End(E_1), End(E_2)$, is it cyclic?

Given a fixed integer $n > 1$, is it possible to classify the isogenies $E_1\rightarrow E_2$ of degree $n$ up to automorphisms of the source or target?

References would be appreciated as well.