What can we learn from the energies of molecular orbitals?
Not a full answer, but some general thoughts:

In general, what we can learn from individual orbitals (and their energies) is very limited, since orbitals are just an abstract tool to represent the many particle electronic wave function. This is manifested by the need for methods beyond Hartree-Fock (DFT, post-HF, etc) and the fact that the orbitals in these methods become even more abstract: Kohn-Sham DFT considers the orbitals to be non-interacting and moves the electron-electron into an external potential. In post-HF the ansatz for wave functions is extended in different (yet similar) ways, but ultimately still based on orbitals. Thus, an exact theory does not consider individual orbitals, but always combines them. In the context of such a theory, it is thus very hard to relate a molecular property to orbital energies only.

However, there still seems to be some rough correspondence, as MO diagrams are widely used to explain things like stability and reactivities. But I think this is all just empirical. Also, the fact that orbitals are an approximation means it does work on some scale. For example, we can distinguish between core and valence electrons based on the orbital energies, as those are typically well separated. (the cutoff energy may however depend on the system). But arguing which valence electron is more strongly bound is more difficult, as their orbital energies are more similar. But this is not really helpful, since we can separate core and valence electrons without applying quantum mechanics.

But we can extend that idea a little bit: Looking at differences in orbital energies, we can estimate the distribution of the electrons and predict a high-spin or low-spin situation of unpaired electrons (e.g. in transition metal compounds). Or the stability of radicals. Or the stability of bonds, e.g. in $\ce{C=C}$ bonds, the $\sigma$ bond is more stable than the $\pi$ bonds. We can even explain the more stable bonding $\ce{N2}$ compared to $\ce{O2}$, although we are comparing two systems with different nuclear charges and number of electrons here. But these arguments mostly work without looking at the actual numbers for orbital energies. It is mostly just qualitative.

Ultimately, we still know something about the orbital energies that we can use. The values are just very fuzzy (high uncertainty, large error bars), therefore we have to be careful with any conclusion. This may actually be the reason for how a large part of chemistry actually works: Often there are concepts which kind of work, but are not applicable in every situation, e.g. the HSAB concept. There may be other competing theories which cover other situations, but a full exact theory (other than elaborate quantum mechanics) rarely exists. In chemistry everything is a bit fuzzy and often it comes down to just know or test the properties of a certain compound. Physics on the other hand can be much more systematic.