There's no such thing as a game specifically for a DS lite. DS games will work on a DSi, as long as it does not require slot-2, like Guitar Hero.
Yes, the Nintendo DS lite can play DSi games
Yes Nintendo ds games work on DS, DS Lite and DSI. The DSI does not support gameboy advance games but the DS does.
All DS games work on the NDSi.
Yes, Ds games work with the Dsi as well, but not Game-Boy games
Yes, you can play DS lite games on the DSi as well as the DSi XL
all types should work.
A DSi uses DS games.
Of course they do, you chickens! :D
There is no such thing as "DSi Lite" but let me tell you this: at the moment of writing, ALL DS games fit in ALL DS's (DS, DS Lite, DSi, DSi XL)
No. Action Replay DS Lite only works on DS Lite. You will have to buy an Action Replay DSi for your DSi.
Gameboy and GBC games do not work on DS or DS lite. Gameboy Advance games can be inserted into slot-2 of the DS and DS lite. However none of them are compatible with the DSi or DSi XL.