As of today, it is not possible. Not even for the DSi. Sure you can use DS Download play using a certain Wii channel to play trials of new games, but you cannot play Wii games on any current version of the DS. There are some titles that are in both the Wii's and DS' library, but that is completely irrevelant.
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You cannot play Wii games on any Nintendo DS system.
Yes, the NDS lite can play all DS games
No. There is no way to download or play DSiWare games on Nintendo DS or DS Lite systems.
It can play GameBoy games and DS games.
Yes, the original Nintendo DS as well as the Nintendo DS lite can play NDSi games.