It doesnt really matter. As long as it is approved by Microsoft (which is 99.9%) then you are in the clear.
i think the answers obvious,YES
yes unless you have the newest xbox 360 (black slimline), If not you have to use either an adapter or a ethernet cable
Xbox Live requires an internet connection, either by wireless adapter or ethernet cable.
You either have to have the wireless adapter or the xbox 360 slim (which is better)
No, you either have to have an Ethernet cable or have a Xbox 360 wireless adapter.
Xbox 360 adapter? You can get a Xbox 360 wireless adapter.
It is highly advised because it will provide you with the best connection for playing with friends.
You need an xbox live gold card and either and ethernet cable or a wireless adapter
I am not going to say it is impossible for a different adapter to work, just unlikely . The Xbox 360 Wireless Network Adapter was made specifically for that purpose, so it is unlikely that a normal adapter will work if attached to the Xbox as they most likely will not be compatible.
new
Xbox Wireless Network Adapter MN740
you can get your xbox 360 ac adapter to work with your xbox 360