Yes, the DSi can connect with the Wii just like an ordinary DS can with game demo downloads, new game features, and Miis in some games.
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The Wii came from Nintendo, just like Nintendo DS's or Nintendo DSi's.
No
No, the Sony PSP cannot connect with the Nintendo Wii
no but you can connect to the wii to trade stuff etc
Yes, the DSi (and XL) use the same memory cards as the Wii does.