To be honest, I do not really understand the second part of your question. For example, why is the map H/G to H/PSL(2,Z) induced by a self map of H? You need this to have the map i:G->PSL(2,Z).

Pete wants his elliptic curve to be semistable. Therefore you need to require that "the E you start with" has an even number of `unstable' fibers and each unstable fiber is of type $I^*_{\nu}$. Such surfaces exist, but it requires a little work to show the existence by given a Weierstrass equation for exmaple. The K3 case is easier since you can just refer to the surjectivity of the period map.