No the wireless signal will not damage your tv.
The XBOX Slim will have free wireless.
A wireless adapter is not included in the standard xbox package.
Xbox Wireless Network Adapter MN740
There is a wireless chip for the Xbox 360 S.
It allows you to connect to xbox live on a wireless network
Yes, your xbox can connect with your wireless router. You must buy the wireless add on for the xbox. If you don't wish to pay $100 for the wireless add on, you can just connect an ethernet cable from router into the xbox.
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
The adapter connects you to Xbox Live but only if you have a wireless network and if you have a Xbox Live subscription.
Yes You Do
An xbox
The Xbox 360 S has a wireless chip which can be replaced.
The Xbox 360 utilizes a built in wireless N adapter that transmits wireless data between the Xbox and your wireless router.