The unit length hypothesis at here is not important, and very crude estimates are available using Sobolev embedding only. The main issue is that studying the spectrum on the manifold itself is not enough for recovering underlying topological/geometric information of the manifold. This is a subtle topic even for 2 dimensional surfaces, where a lot of work has been done.
For very recent work, check some papers by Sogge and Xi:
https://arxiv.org/abs/1711.04707
I would suggest that instead of working through the detailed estimates (on a sphere, on a torus, on negatively curved manifold, etc), think about some other ways to understand the spectrum of the Laplacian on the manifold. For example, a compact Riemann surface of genus $g\ge 2$ can be realized as the quotient of the upper half plane $\mathcal{H}/\Gamma$. There is a lot of interesting work that can be done to understand the relationship between the group action and the spectrum. The interplay between the algebraic nature of the surfaces and flexibility of the analysis tools made the subject really interesting.
A survey paper by Sanark may be a good start:
http://web.math.princeton.edu/facultypapers/sarnak/baltimore.pdf
For 3-manifolds this becomes deep and is related to heat kernels in geometric analysis. The subject is related to Ricci flow and there are plenty written up online already.
I would suggest against choosing this as a thesis research topic (you wrote "The major focus of the research that my advisor has me doing..."), and I encourage you talk to your advisor on this.