You need to connect to the DSi store, and pay for them there. You can then download them to your DSi.
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No. There is no way to download or play DSiWare games on Nintendo DS or DS Lite systems.
Yes, but the only DSiWare that is available for free as of now is the Nintendo DSi Browser and Flipnote Studio.
Mario vs. Donkey Kong: Minis March Again!, Shantae: Risky's Revenge, and Cave Story are among the best DSiWare games available. Really, though, it depends on your opinion.
For the most part, yes you can. If you want to connect with pictochat or DS games (including DS Download), you can connect between a Nintendo DS and DSi. However, for DSi specific games, such as DSi games and certain applications, the Nintendo DS will not be able to connect. But, note that certain DSiWare games that can DS download play still can be connected to the Nintendo DS.
I have just been lookin on the Nintendo website and you can play your DSiware games straight from the SD Card + you can copy DSiware game save files to your SD card. You can't however transfer your DS games to an SD card or transfer your DS game save files to your SD.