Eevee quest question

No; the Weather (currently) has no effect on what the Eevee will evolve into.

Yes, getting the Espeon is about attention to detail. Walk at least 10 km, get at least 2 candies as buddy, and be buddied with him/her when evolving. Best strategy is to buddy up and go 10km continuously. After that you can change buddies, but when you want to evolve to Espeon (day) or Umbreon (night), make sure you buddy up with him/her again, and that you have a good signal, so the game knows where you are, and more importantly, when you are.