At the risk of blowing my own trumpet, I feel like I ought to mention a recent preprint of mine that addresses this question.

Marc Palm answered your question 2, 3, and 4 reasonably well. (For question 4, the reference for where it is published is Cartier's paper.) For question 1, I know that Bolte and Johansson certainly expect the conjecture to be true for $\Gamma_0(q) \backslash \mathbb{H}$ with $q$ squarefree, provided one first removes all the Maaß oldforms (those coming from lower level), as they will always give rise to spectral multiplicity at least $2$. So for $q > 1$, the conjecture should be modified to only be about the eigenvalues of Maaß newforms.

Bolte and Johansson (and later Strömbergsson) describe a spectral correspondence between the eigenvalues of Maaß newforms on $\Gamma_0(q) \backslash \mathbb{H}$, $q$ squarefree, and eigenvalues of the automorphic Laplacian for the group of units of norm one in a maximal order in an indefinite quaternion division algebra over $\mathbb{Q}$. Bolte and Johansson conjecture that the spectrum of this automorphic Laplacian is simple (see the Hypothesis on p.61), and hence that the eigenvalues of Maaß newforms on $\Gamma_0(q) \backslash \mathbb{H}$ are simple when $q$ is squarefree. Here we are looking at the congruence subgroups
\[\Gamma_0(q) = \left\{\gamma \in \mathrm{SL}_2(\mathbb{Z}) : \gamma \equiv \begin{pmatrix} * & * \\\ 0 & * \end{pmatrix} \pmod{q}\right\},\]
\[\Gamma_1(q) = \left\{\gamma \in \mathrm{SL}_2(\mathbb{Z}) : \gamma \equiv \begin{pmatrix} 1 & * \\\ 0 & 1 \end{pmatrix} \pmod{q}\right\},\]
\[\Gamma(q) = \left\{\gamma \in \mathrm{SL}_2(\mathbb{Z}) : \gamma \equiv \begin{pmatrix} 1 & 0 \\\ 0 & 1 \end{pmatrix} \pmod{q}\right\}.\]

On the other hand, I show in my preprint that if $q$ is odd but not squarefree, then the new part of the spectrum of the Laplacian on $\Gamma_0(q) \backslash \mathbb{H}$ is *never* simple; there is always a positive proportion of eigenvalues $\lambda \leq T$ for which the corresponding eigenspaces of Maaß newforms are at least two-dimensional. For $\Gamma_1(q) \backslash \mathbb{H}$, the situation is even worse: there is spectral multiplicity even if $q$ is squarefree (provided $q \neq \{1,2,3,6\}$), and the dimension of an eigenspace can be proven to grow with $m$ if $q = p^m$ for an odd prime $p$. The proof doesn't use anything about the eigenvalue $1/4$, as in GH from MO's answer, but rather looks at twists of newforms that have the same level $q$ after twisting; this gives rise to spectral multiplicity.

There is one remaining case where I know that spectral multiplicity must occur; that of $\Gamma(p) \backslash \mathbb{H}$, with $p$ an odd prime; Randol shows that there must be infinitely many eigenvalues with multiplicity at least $\frac{1}{2} \left(p + (-1)^{(p - 1)/2}\right)$. I don't think anything is known for noncongruence subgroups.

I still think that for any congruence subgroup $\Gamma$, the multiplicity of an eigenvalue of the Laplacian on $\Gamma \backslash \mathbb{H}$ ought to be uniformly bounded as $\lambda \to \infty$, with the bound depending on $\Gamma$. Unfortunately our only tool for attacking this question (finding an upper bound on the multiplicity) seems to be via getting better error terms in the Weyl law, and we are very far off making that approach be useful.