Much has been said about bounds on Laplacian eigenvalues, and the literature can be tough to sort through! I am specifically interested in the case where the domain is a closed compact surface, and am seeking bounds on the difference between the two smallest (in magnitude) distinct nonzero eigenvalues. Any good pointers are greatly appreciated!

The question is a bit vague, but I'm assuming you're asking for an upper bound on the gap between the smallest nonzero eigenvalue and the second smallest nonzero eigenvalue (ignoring multiplicity), let's call this number $E$, taking the supremum over all closed hyperbolic surfaces. If $\lambda_j$ is the $j$th eigenvalue, counted with multiplicity, then Otal has shown that $\lambda_{2g2}>\frac14$. Moreover, Buser has shown that this estimate is sharp: one can find for any $\epsilon >0$ a surface with $\lambda_{2g3}<\epsilon$. Now, I believe that in these Buser examples, one could actually arrange that $\lambda_1=\lambda_2=\cdots=\lambda_{2g3}$, by adjusting the moduli (recall moduli space has dimension $3g3$). If so, this would show that $E\geq \frac14$. On the other hand, a result of Besson implies the maximal multiplicity of $\lambda_1$ is $4g+3$, so we know that $E\leq \lambda_{4g+4}$. There is some universal bound on $\lambda_{4g+4}$, since by the Margulis Lemma, there exists $R>0$ so that on a Riemann surface of genus $g$, one can find $4g+5$ disjointly embedded disks of radius $R$ (there is such an $R$ for any linear function of $g$). By Corollary 4.65 of GallotHulinLafontaine, one has $\lambda_{4g+4} \leq \lambda_1^D(D_{R})$, where $\lambda_1^D(D_R)$ is the first Dirichlet eigenvalue on a hyperbolic disk of radius $R$. Thus, $E\leq \lambda_1^D(D_R)$. One ought to be able to get explicit bounds on $R$ and $\lambda_1^D(D_R)$ using computations of Margulis constants to get an estimate of $E$. It is conjectured by Colbois and Colin de Verdiere that the maximal multiplicity of $\lambda_1$ equals the chromatic number of a genus $g$ surface, which grows like $\sqrt{g}$. Given Huber's result that $\sup \lambda_1 \to \frac14$ as $g\to \infty$, this would imply that $E_g$ is asymptotic to $\frac14$, where $E_g$ is the maximal difference between these eigenvalues for genus $g$ hyperbolic surfaces. 


I think that in general $vol(X)$ is the best possible upper bound for the multiplicity of the first nontrivial eigenvalue. This can be proved rigorously for compact Riemann surfaces with the Selberg trace formula, but I would guess this holds more general via an analysis related to Weyl laws. For certain arithmetic compact Riemannian surfaces (associated to division algebras), the first nontrivial eigenvalue is assumed to be larger or equal $1/4$. Some better upper bounds in this particular case are due to Michel and Venkatesh (the JacquetLanglands correspondence has to be applied). http://math.stanford.edu/~akshay/research/MV.pdf The Langlands correspondence also suggests that the multiplicity can be arbitrary large. 

