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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.
You don't download them on a lite, you'd need at least a DSi.
If you mean by Wi-Fi, you need to have it and go to the DSi Shop Channel.
Yes, the original Nintendo DS as well as the Nintendo DS lite can play NDSi games.
DSi Ware games and other downloadable content The Nintendo DSi Shop is specific to the region for which the system was manufactured. For example, a Japanese system can download DSi Ware games and other software only from the Nintendo DSi Shop for the Japanese region, regardless of where the system is being played. Nintendo DSi exclusive game cards Nintendo DSi game cards that are only compatible with Nintendo DSi and Nintendo DSi XL (and not Nintendo DS or Nintendo DS Lite) are region locked. This means game cards manufactured for the U.S. market would not work in a system made for another country. Nintendo DS / Nintendo DS Lite game cards Older Nintendo DS and DS Lite software is region-free, so you can play most of that software on a Nintendo DSi and Nintendo DSi XL from any region. It's possible, however, that future Nintendo DS software will incorporate region encoding. In these instances, these games would not work in a system made for another country. Source: http://www.nintendo.com/consumer/systems/dsi/en_na/faq.jsp#vers