Yes you can. You can also connect DSi to DS and DS Lite to DSi.
yes
no but you can buy an accessory for the gamecube that lets you play game boy games on.
No, you can not connect the DSi with the DS lite. you can with picto chat
Yes. However you will need a game card that supports multi player. You connect wirelessly to other ds systems nearby. Sometimes you can connect with just one game card. However usually each player...