No, you CANNOT evolve you're partner Eevee. It refuses to. Howver, you can evolve a different eevee using the water , thunder, or fire stone. You can find one in Celadon City or Route 17
Source: Knowledge
It is impossible to evolve your starter Eevee. It is also impossible for it to be shiny. Eevee that you find in the wild can evolve, however. Eevee is a wild encounter on Route 17
Source: My knowledge of playing the game
No. Just the same as Let's Go Pikachu, Eevee and Pikachu is not able to evolve.
