Zevahim 101: Moshe, Aharon, and Inaugurating the Mishkan

The Gemara addresses the acute mourning of Aharon (Aaron) in the Torah when Nadav and Avihu die on the altar, when they bring the "strange fire" - and, among other offerings, a sin-offering was brought too - and Moshe (Moses) rebukes Aharon for burning the sin-offering among them. How much does Aharon's conduct inform the practices-to-be of kohanim in a state of acute mourning in the generations to come? Also, was Moshe himself a kohen, given his partaking of the offerings on this day of establishing the Mishkan? Doesn't the very fact that he was able to eat from the offerings mean he must have been a kohen? It's not that simple