PAK really isn't on DS games, It's on the DS menu. On a normal DS or DS lite, the bottom item on the menu (Excluding the settings button) should say "Game PAK not inserted" or something like that. It basically means there is no game boy game inserted. Game PAK really is a Game boy cartridge.
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The rumble pak is an accessory that fits into the slot-2/GBA slot of a regular DS or DS lite. Some DS games will recognise it and it makes the DS vibrate, say when you get hit by something in a game.
that means you have something in the gba slot but its for the ds games and you cant play it by itself. for example if you get guitar hero for the ds and put the guitar controls in the gba slot it will say ds option pak inserted
you put a game pak in the back of the system and you will be able to play with touch sensitive screen.
with an r4 you can. you can download leaf green onto the r4 then play it through the ds slot.
You can't delete a game, as they are on the game cards. If you mean the data, you should be able to delete it in the options menu of that specific game.