The Wii wants to do a system update, but isn't letting you connect because your Wii is not updated. Catch 22.... Anyway, go to the connection settings it prompts you to, choose your connection one, save, then test. Should work now, and will prompt you to do a system update.
Depends on how far you are from your router and the construction of walls . You may need to get an extender which acts as a relay place it somewhere between the original router and the downstairs location. If it is a smart tv, follow the directions for the tv and make sure the adapter in the tv is compatible with your router. Check the specs.