You can play any normal Nintendo DS games on a DS Lite but they do not make separate games for the DS Lite. The DS Lite can also play any Game Boy Advance (not colour or original) game but the game protrudes from the DS Lite which makes it look slightly stupid.
Yes, you can use ds lite games in the original Nintendo ds. They are not making new specialized games for the Nintendo ds lite. they are still making games that fit both the original DS and the Nintendo DS Lite.