For congruence subgroups of $PSL(2,\mathbb{Z})$, the Weyl law for the eigenvalues of Maass cusp forms had been proven by Selberg. How is the status of such a Weyl law for eigenvalues of Maass cusp forms for arbitrary arithmetic nonuniform cofinite Fuchsian groups? Has it already been proven? Does one know at least that there are infinitely many linearly independent Maass cusp forms for these lattices?

The Weyl law has been proven in great generality, e.g. for congruence subgroups of $PGL(n,\mathbb{R})$, and beyond. See the introduction in LindenstraussVenkatesh: Existence and Weyl's law for spherical cusp forms, GAFA 17 (2007), 220251 (manuscript available here). See also Marc Palm's recent thesis. 


Consider the normalizer $\Gamma$ of $\Gamma_0(N)$ in $SL_2(\mathbb{Q})$ for $N$ squarefree, which is not a subgroup of $SL_2(\mathbb{Z})$, then you obtain the same Weyl law as classical. For my taste, these groups should also be called congruence subgroups, although they are not contained in $SL_2(\mathbb{Z})$. They are important if you want classically distinguish between nonisomorphic supercuspidal reps contained as factors of automorphic reps associated, which are associated to ramified quadratic extensions of $\mathbb{Q}_p$ at $p N$. I don't know a single example what happens if $\Gamma$ does not contain a congruence subgroup, but is arithmetic. Btw, your question makes perfect sense, because the modified RoleckeSelberg conjecture states (see the introduction http://link.springer.com/content/pdf/10.1007%2FBF02572621.pdf) that apart from arithmetic lattices, there are at most finitely many discrete eigenvalues. There is some computational evidence by Hejhal for this conjecture: Hejhal, Dennis A.(1MNSM) On eigenvalues of the Laplacian for Hecke triangle groups.
What is sure is that if $\Gamma$ does not contain a congruence subgroup, then the Maass cusps forms on $\Gamma \backslash \mathbb{H}$ can not be lifted to a vector of an automorphic adelic space $SL_2(\mathbb{A})$ in any obviuous way, and one cannot expect them to have a nice $L$functions. Also the contribution of the continuous spectrum is most likely not expressable in terms of classical Dirichlet Lfunctions. Recall that this is the main point to be understood for Weyl laws of nonuniform lattices. I know that the Belyi's theorem generates an isomorphisms between nonsingular algebraic curves over $\mathbb{C}$ and $\Gamma \backslash \mathbb{H}$ compactified at the cusps, where $\Gamma$ is an arithmetic subgroup of $SL_2(\mathbb{Z})$, but noncongruence in general. This might be a source of such lattices and at least modular functions on them should exists (by Riemann Roch?). 

