Answer to 1. If you have your PC connected to the internet via a wireless router, then no, you don't have to have it on to have wireless internet; I use my laptop all the time while our desktop PC is turned off.Originally Posted by stu
Answer to 2. As far as I understand it, Wii will connect to the internet just as the DS does; all it needs is a wireless internet connection, or a usb-internet adapter if you don't have a wireless router; a wired connection would also most likely be faster/more reliable. I'm guessing the routers that are compatible with Wi-Fi for the DS, which can be found here, will be compatible with Wii.
Hope this answers your questions