The following is an attempt to formulate a couple of questions which have been lurking in the back of my mind for a while. I'm sorry if this is long, or if my terminology is not correct, or if my notation is non-standard, or if it turns out that I didn't understand a single thing that I thought I understood.
Suppose we are given a genus $0$ congruence subgroup $G$ of $\Gamma=PSL_2(\mathbb{Z})$.
The (compactified) modular surface $X_G={\mathbb{H}/G}$ is then a degree $n=|\Gamma:G|$ covering of $X=\mathbb{H}/\Gamma$. Let K'=\mathbb{C}(w) denote the function field of $X_G$ and $K=\mathbb{C}(j)$ denote the function field of $X$. If $G$ is normal in $\Gamma$, the field extension K'/K is a Galois extension with Galois group $\Gamma/G$. Klein's $j$ function can be written as a rational function in $w$, of degree $n$.
Now each elliptic curve $E/\mathbb{C}$ corresponds to a point of $X$. We can write such a curve explicitly as the locus of zeroes of a cubic in $\mathbb{CP}^2$; its coefficients are rational in $j$. For almost all choices of $E$, there are $n$ points of $X_G$ lying over $E$. In some cases, these points are known to correspond to some pairs $(E, S)$, where $S$ is some sort of substructure of $E$ (for example a "level structure").
What's not completely obvious to me is that such pairs $(E,S)$ should also be given by a cubic or quadric equation in normal form, whose coefficients are rational in $w$. Here's an example. The group $\Gamma(2)$ is normal in $\Gamma$; the quotient $\Gamma/\Gamma(2)$ is isomorphic to $S_3$. The projection map $X_{\Gamma(2)} \to X$ is a six-sheeted ramified covering. The generator for K' is then the modular function $\lambda$. The points of $X_{\Gamma(2)}$ correspond to pairs $(E, S)$, where $S$ is a basis (over $\mathbb{F}_2$) for the $2$-torsion subgroup of $E$. Since this group is isomorphic to the Klein four-group, there are indeed $6$ different choices of bases for it over $\mathbb{F}_2$.
Now the fascinating thing is that we can consider such a curve as given by an equation in Legendre normal form $y^2=z(z-1)(z-\lambda)$. For most choices of $\lambda$, there are precisely six curves, in this form, which are isomorphic to this one; the other corresponding values of $\lambda$ are related by the action of $\Gamma/\Gamma(2)$.
Now my first question is: how would one "discover" the normal form $y^2=z(z-1)(z-\lambda)$, given only the group $\Gamma(2)$? Assume, perhaps, that we know (at least) the normal form associated to $j$. Given any subgroup $G$ of $\Gamma$, can we find a normal form associated to $G$ in this manner? If the genus of $X_G$ is bigger than $0$, can we hope, perhaps, to find a similar construction?
My second question is a bit more vague: what are other modular surfaces moduli spaces for? This is understood in some specific cases, but is a general answer to be hoped for? Are the examples which are understood well understood? Can we hope to associate to each modular surface a normal form, in which the extra "structure" $S$ (whatever it may be) is encoded?
Anyways, thanks for reading, and I hope that this made sense!