No. The NDS lite cannot play 3DS games in the first place so it will not be able to link with a 3DS.
However, the 3DS is backwards compatible, meaning it'll play NDS games so it can link with the DS lite if playing a NDS game.
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No, you need a 3DS at least.
Yes, Pokémon Platinum was created to be a Nintendo DS game so it can be played on all of the Nintendo DS systems which includes the original Nintendo DS, the Nintendo DS Lite, the Nintendo DSi, the Nintendo DSi XL, the Nintendo 3DS and the upcoming Nintendo 3DS XL.
As of February 2014, there are 7 versions of the DS. DS and DS lite DSi and DSi XL 3DS, 3DS XL and 2DS
You can't 'add/download' games to the DS lite.
There is no such thing as a 'DS lite' game. If it's a Nintendo DS game, it works on any model.