Yes, they can fit.
DS/DS Lite Slot-1 games are exactly the same as DSi Slot-1 games, but Gameboy Advance (GBA) games are not supported by the console.
Important:
Any Nintendo DS game which requires an addon that uses the GBA slot, is not supported on the Nintendo DSi (eg. DS Lite internet browser or Guitar Hero: On Tour)
Note:
Because of the above reason, do not sell your Nintendo DS Lite if you intend to play Guitar Hero On Tour. :)
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The DS lite and DSi are completely different pieces of hardware. You would have to buy a DSi, you can not turn a DS lite into a DSi
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
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.
its an Internet browser for the Nintendo DS Lite (you can also get aversion for NintendoDS) it is integrated in the Nintendo DSi
Yes, the original Nintendo DS as well as the Nintendo DS lite can play NDSi games.