I don't believe any of the game boy color games are combatible with the ds light. I'd suggest an sp, gba, or gbc if you want to play any of the old games.
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Yes, Pokemon SoulSilver is a Nintendo DS game so it can be played on a Nintendo DS
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.
Sadly, you are unable to, You will need to 'transer' the pokemon from the GBA game in the Nintendo DS fat/lite.
Trading between an DS and a DS lite is the same as between a DS and a DS. There is no difference.
You can trade between two DS but only with DS games, otherwise you'll have to migrate them but then you cant put them back on a GBA game anymore.