Is there any advantage of evolving an Evee or any other Pokemon at a certain level using the proper elemental stone?
You can evolve a stone pokemon at any level. However, they tend to not learn any worthwhile moves after evolution, so make sure you get all the moves you want before evolving by leveling up. TM/HM moves, move tutor and egg moves work well too.
I would say no. But there are advantages because the pre-evolution may learn moves that the next evolution state won't. It varies on the pokemon it evolves into like Pikachu evolving into a Raichu. Level Pikachu as high as the last move on its move set because Raichu doesn't learn any moves. ^^
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