For your first question, I can give an application close to my own field. Perhaps a more Langlands-ish person can say something interesting about applications to modularity of Galois representations.

The moduli interpretation of level structures can appear naturally in orbifold 2D conformal field theory. In ordinary 2D CFT, one has a notion of correlation function, that takes as input a Riemann surface together with some data decorating closed submanifolds, and outputs a number. For elliptic curves with trivial decorations, one then obtains modular functions. In orbifold CFT, the input data is upgraded to a Riemann surface with a branched $G$-cover, for $G$ a finite group. In this case, for elliptic curves with trivial decorations, one obtains functions on moduli spaces of elliptic curves with $G$-torsors.

The simplest non-trivial case is for $G$ a cyclic group of order $N$. If we ignore the problem of fixing a basepoint, a $G$-torsor over an elliptic curve $E$ is a disjoint union of $N/k$ isomorphic elliptic curves, each with a cyclic $k$-isogeny to $E$ whose kernel has a distinguished generator, related by a cyclic group of isomorphisms, for some $k$ dividing $N$. Keeping this in mind, it is not hard to show that the (coarse) moduli space is a disjoint union of $Y_1(k)$ as $k$ ranges over divisors of $N$. In other words, correlation functions in this setting are given by a list of modular functions whose levels are divisors of $N$.

This particular family of examples has come up in my own work, because there is a natural notion of Hecke operator $T_n$ on these functions. The condition that $nT_nf$ is a monic polynomial in a function $f$ for lots of $n$ implies the restriction of $f$ to any component generates the function field for some genus zero quotient of $\mathcal{H}$. The monic polynomial condition then allows for the construction of holomorphic infinite products as Borcherds-Harvey-Moore lifts, and makes the analysis of some closely related infinite dimensional Lie algebras tractable.

For your second question, I vaguely recall hearing about some work of Scholl to the effect that non-congruence forms have some connections to automorphic forms for higher-rank groups through their L-functions. Bold guesswork might suggest that non-congruence curves arise from natural moduli problems related to higher-rank motives, but presumably less straightforwardly than the way Shimura curves are moduli spaces of abelian surfaces with quaternionic multiplication.