Just to the left of the Pokemon center in Opelucid City. It's called the Battle House in Pokemon Black and White.
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No they can't.
black and white
No. Pokemon Black and White have new Pokemon that the past games' engines cannot recognize. Even if you battle with Pokemon from previous generations, they might be loaded with moves from the new generation.
yes,you can battle with any Pokemon game that's for the ds/dsi/the new 3DS.
You cannot re-battle the Gym Leaders in Pokémon Black and White.