Okay, so I've found out that Eevee can be found in Castelia City as a gift. Um...where? Anyone know who I have to find or what I have to do in order to get this?

You must beat the game and talk to Amanita to receive the Dream World ability Eevee. Basically, no free Eevees! Also, the location is at the far left of Castelia in the building across from the gym. (Far left, I know that much)
Well, Amanita or Fennel. They're always together, so it's one or the other.

There is also a 5% chance of getting it in the Castelia Park. Which is southwest of the entrance of the Castelia Sewers. The park looks like a PokeBall with a tree in the middle.

After you gain access to the Castelia Sewers, look around for a stairway that isn't the one you came in from. If you go up that stairway, you'll come up into an isolated little part of Castelia city with a Poké Ball-shaped pattern of grass. There's a 5% chance of finding an Eevee in that grass. (Until you clear Team Plasma out of there, there'll be a person there who blocks access to this stairway, so be sure to do that first )