There may be more than these, but it will probably only be between similar games.
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Nintendo makes the Wii and DS. No offence, but it does say on the product and packaging box Nintendo Wii and Nintendo DS.
On the main menu screen, select "Roadie Play", choose whatever way you want to play, and follow the on-screen instructions. Then on your Nintendo DS, choose DS Download Play and wait for the DS and Wii to connect.
The Wii came from Nintendo, just like Nintendo DS's or Nintendo DSi's.
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.
Yes, it connects to other Nintendo DS's, the Nintendo Wii, and to the internet with it.