If you want baby Eevees, you'll need either Ditto, or a female Eevee/Eeveelution.

The schedule for the Royal Unova is the following:

Winter: 5 PM to 7 PM.
Spring and Autumn: 5 PM to 8 PM.
Summer: 7 PM to 9 PM.

If you're post-game, then you can also battle Hugh once a day in Driftveil (at the house with the former Team Plasma members).

You can also battle Morimoto and Nishino at the GameFreak building in Castellia.

Also, during spring, you can battle Cynthia in Undella Town.

There is also the White Treehollow, which can give your Pokemon some good training even at high levels (if you can put up with not healing while you're in there).

And you can go to the restaurant in Striaton, where you can Side with either Chili, Cilan or Cress for a double-battle against the other 2 brothers.