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
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.