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He was a genius and naturally discovering advanced hidden Hyuga techniques at a very young age. Seems likely to me.
That was in Part 1. But in part 2, with the emergence of haxxed abilities such as Mangekyō Sharingan, Rinnegan, Izanagi & Izanami, Jūbi's power ( Rikudō Senjutsu ), Neji later went on to become a fodder. I'm pretty sure than Hamura possessed a unique set of abilities besides the Byakugan in his repertoire of techniques.
It's possible, and it'll probably happen just so Neji doesn't stay dead.
I even forgot about Neji and the people at the now destroyed headquarters, lel
As far as we can telll hagoromo did not have a transmigrant at all. I would think that is the main reason he picked naruto and sasuke to split his powers. I doubt either could really work as a transmigrant, neither has the innate potential for his complete powers. Sasuke only has the potential for the rinnengan and naruto only has the potential for his chakra. Neji in turn only has the byakugan, a power which hagoromo altogether did not have. If neji was to be the transmigrant of anything it would have to be hamura whose genes neji presumably inherited.
Kinda sad the story wasn't diverse enough to include other characters (actual characters developed and made relevant for the story, connected to the Kaguya)