  The $A$-site spinel NiRh$_2$O$_4$ is the only known realization of a spin-1
diamond lattice magnet and is predicted to host unconventional magnetic
phenomena driven by frustrated nearest and next-nearest neighbor exchange as
well as orbital degeneracy. Previous works found no sign of magnetic order but
found a gapped dispersive magnetic excitation indicating a possible valence
bond magnetic ground state. However, the presence of many competing low energy
degrees of freedom and limited empirical microscopic constraints complicates
further analysis. Here, we carry out resonant inelastic x-ray scattering (RIXS)
and x-ray absorption spectroscopy (XAS) to characterize the local electronic
structure of NiRh$_2$O$_4$. The RIXS data can be partly described by a
single-ion model for tetrahedrally coordinated Ni$^{2+}$ and indicates a
tetragonal distortion $\Delta t_2\!=\!70$ meV that splits the $t_2$ orbitals
into a high energy orbital singlet and lower energy orbital doublet. We
identify features of the RIXS spectra that are consistent with a Rh-Ni two-site
excitation indicating strong metal-metal hybridization mediated by oxygen in
NiRh$_2$O$_4$. We also identify signatures of electron-phonon coupling through
the appearance of phonon sidebands that dress crystal field excitations. These
results establish the key energy scales relevant to the magnetism in
NiRh$_2$O$_4$ and further demonstrate that covalency and lattice dynamics play
essential roles in controlling the magnetic ground states of $A$-site spinels.
