Xbox 360S has a built in Wi-Fi adapter
Either use a router or wireless internet.
You can use a wireless connection if you have a wireless router
a wireless modem means it has a wireless router built in. A wireless router means just a wireless router. if your modem is not wireless, you will need just a wireless router
It does not matter if you are connected through wireless or cable, your router can provide firewall protection for all of its ports on the LAN side. If your question is: Is my wireless router's firewall protecting me, although I do use it for wireless connectivity, the answer is yes. Your routers firewall protects you whether you use the wireless option or not.
Yes. All you need to do is re-sync it to their Xbox360.
Your router WILL need to be wireless to connect wirelessly to your Xbox, but you can use an ethernet cable if your router is not wireless, straight from the router to the XBOX
You can use router of G or N type and the same type wireless adapter. Of course you can use wireless N adapter with G-type router and reverse.
yes
You can use your Xbox360 Wireless headset to play!
802.11
Yes.
there has been a misunderstanding between modem and router. some people use only a modem to connect wirelessly and some use a modem and a router. The fact is there are two types. first type is a modem with built in router, second is router itself. this will clear your mind. you don't have to buy a router to go online, just a get modem with router built in