We present a detailed structural investigation of self-assembled indium gallium nitride nanodots grown on c-plane aluminum nitride templates by the droplet heteroepitaxy technique in a plasma-assisted molecular beam epitaxy reactor. Various growth parameters, including the total coverage of the metal species, relative and total metal effusion fluxes, and nitridation temperature were investigated. Analyses of in situ reflection high-energy electron diffraction patterns and comparison with simulations showed that the resulting crystal structure was a mixture of wurtzite and twinned zinc blende phases, with the zinc blende phase increasingly dominant for lower metal coverages and lower nitridation temperatures, and the wurtzite phase increasingly dominant for higher nitridation temperature. Studies by field emission scanning electron microscopy and atomic force microscopy revealed that the nanodots exhibit trimodal size distributions, with the dot morphologies of the intermediate size mode often resembling aggregations of distinct clusters. Nanodots grown at higher nitridation temperatures had larger inter-dot spacings, with hexagonal in-plane ordering observable at a sufficiently high temperature. Using grazing incidence small angle X-ray scattering, we determined the nanodots to be approximately truncated cone shaped, and extracted the mean radius, height, and inter-dot distance for each distribution. Microstructural investigations of the nanodots by cross-sectional transmission electron microscopy indicated that the majority of the dots were formed in dislocation-free regions, and confirmed that the intermediate size dots were approximately truncated cone shaped and consisted of both zinc blende and wurtzite regions. Mapping of the elemental distributions by energy dispersive X-ray spectroscopy in scanning transmission electron microscopy mode indicated highly nonuniform indium distributions within both small and intermediate size dots which are potentially indicative of indium clustering and kinetically controlled nanoscale phase separation rather than the spinodal decomposition associated with bulk diffusion. The observed enrichment in indium concentration towards the tops of the nanodot layers could be ascribed to the compositional pulling effect.