Yes. Eventually you will acquire 3 stones which can be used to evolve Eevee in to Flareon with the Fire Stone, Vaporeon with the Water Stone, and Jolteon with the Thunder Stone.
Also, according to Unrecognizable and Genga, you can evolve Eevee in to Umbreon or Espeon with happiness and the scenery. Guess I let it faint too often xD. And if the pattern follows, Glaceon in an ice map, and Leafeon in a forest map.
Source: Personal experience. and other users.
Yes, you can, and into all 7 evolutions as well. Flareon/Jolteon/Vaporeon are through the use of stones like normal that you can buy once a special shop comes to you. Umbreon/Espeon/Leafeon/Glaceon are a bit different, some of them evolve once their stats hit a certain number (ex. Eevee has to have a minimum 94ish defense and then finish a battle in the grass city to evolve into Leafeon), others depend on the link % (ex. Eevee has to have I think around 70-80% and then finish a battle in the psychic city to evolve to Espeon. Don't quote me on the exact numbers).
Right now I have a Vaporeon and Flareon, Hoping that eventually enough Eevee will show up for me to get more!! I would strongly suggest getting Vaporeon :D