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.
If you use router to redistribute internet for more than one user, and you want to connect using wireless yes you need both.
You need to have two things: a wireless router and a wireless adapter compatible with the router.
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't. You will need to purchase a new router.
If you are talking about building of wireless network. Then you need a wireless router and a wireless network adapter (if you have a laptop you might not need it). If you have a desktop you rather will have to purchase a wireless network adapter. Pay attention to what kind of wireless network you are going to build. It can be a, b, g, and n types.
If you want to use internet wirelessly, yes you need a wireless router and a wireless adapter for your PC (usually laptops have it).
An ethernet cord to connect the XBox to a router or the wireless adapter for XBox to connect to your wireless router.
You will need a wireless router and a wireless adapter to have wireless pc to pc connection
You need an Xbox wireless adapter.
For such purposes you need at least two subnetworks. One adapter cannot be in the same time in both of them. Thus you cannot do that. If you connect your PC through wireless to another PC, and that PC is connect to the router with a LAN cable that it's doable.
It's not about computer, it's about which wireless adapter you are going to use. If you want take advantage of Wireless-N, you have to have a wireless adapter N.
You need a Wireless adapter, or a ethernet cable connecting to your home router.