The DSi can't execute code on the SD card, all it can do is store pictures on it, or play music. There's no way to play a homebrew game off the SD card, the DSi simply wasn't made with the means to do so.
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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.
go to sound on the dsi menu and select play with music on sd card
You don't need any additional software, just purchase and download games from the DSi shop.
No, the DSi is only compatible with DSiWare.
The dsi sd card is about a inch long, about a 6th of a inch wide.. the sd card made specially for the dsi (the one I have) is white and says Nintendo dsi on it