Well besides the wireless router which provides Wi-Fi, there is no way to connect to the world on your DS. However most coffee places, like Starbucks and Caribou Coffee, have Wi-Fi for free.
They connect in exactly the same way as if it were two regular DS.
i suppose you would connect the same way like tow ds' or two wii's, but i don't think that you can connect a ds and a wii.
There is no way to connect a Nintendo DS to a gameboy SP.
Yes by using ds download Play
No, because there is no way to connect a gamecube to a ds.
Nothing.
yes
It doesn't connect because another opponent isn't in DS download or the opponent is far than 30 feet.
by turning on the computer
No, there is no (official) way to connect the DS to a laptop. With special homebrew hardware, you can connect directly to your DS but this isn't used much. Usually, there are memory cards which can plug into a special adapter (called a flash cart) which you can use to access files on your DS.
It is impossible to do so, unfortunately.
There is no way to connect any DS system to any Game Boy systems.