No, your DS does not need to be on in order to use the PokéWalker however your DS does need to be on in order to transfer a Pokémon into the PokéWalker.
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You do not need a Pokewalker to trade Pokemon. The DS has built-in wireless communications. The DS does not need the Pokewalker to trade Pokemon.
You exit out of the pokewalker and then go to the ds and click the pokewlker menu and click somrewhere where it says gifts and use the pokewalker to transfer to the ds
Take the plastic strip out of the PokeWalker and hold the Center Button for a few seconds. A symbol should pop up. To actually activate the PokeWalker, turn on your DS, choose "connect to PokeWalker" in the menu before accessing your save file. Choose a Pokemon, and put the Pokewalker right next to the DS Game Card. When the DS is trying to connect to it, hold the center button for a few seconds and the transfer will occur. (Make sure there are batteries in the PokeWalker first).
When you start soul sliver press pokewalker and choose you Pokemon and connect it to you ds then Wala!
No, the Pokéwalker can only communicate with Nintendo DS game cartridges for Pokémon HeartGold Version and Pokémon SoulSilver Version via Infrared signals (which can't be picked up by any other devices)