Yes, you can. Some games have a "DS Download Play" Multiplayer feature, where you can play with multiple people (multiple DS's) but with the use of only one DS Game Card. But, some games require two Game Cards. You may also connect to Nintendo Wi-Fi Connection, and Play with DS Owners from around the World.
Yes you can. You can also connect DS lite to DS and DS Lite to DSi.
Yes you can. You can also connect DSi to DS and DS Lite to DSi.
There is no way to connect a Nintendo DS to a gameboy SP.
You cannot connect the Nintendo DS to the Mii channel
For the most part, yes you can. If you want to connect with pictochat or DS games (including DS Download), you can connect between a Nintendo DS and DSi. However, for DSi specific games, such as DSi games and certain applications, the Nintendo DS will not be able to connect. But, note that certain DSiWare games that can DS download play still can be connected to the Nintendo DS.
They connect in exactly the same way as if it were two regular DS.
No, Mario Kart can only connect to other DS systems.
It must be impossible to do this but you can connect a Nintendo DS or DS lite system to a TV
You can't.
no
Yes by using ds download Play
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.