The 'local' connection is when it communicates with another 3DS system. Just keep the devices close together.
You can't watch TV on your 3DS.
Yes, the Nintendo 3DS can connect to the internet
update your Nintendo 3Ds
Yes, it can.
Your wii may not be updated because, when you go to your Mii channel it should have an option on the bottom right corner that says "connect to DS." See the related link.
The closest equivalent would be your 3DS Friend Code.
The 3DS and the Circle Pad Pro connect to each other through the IR (Infra Red) Port on the top of the bottom half of the system. That's why it need batteries.
Yes
You will have to perform a system update, then yes, it does (as long as you have a wireless connection that the 3DS can connect to).
Yes. The 3DS can play DS multiplayer games with any DS.
you can most likely do this but not with the usb lead you connect wirelessly
No you just need an internet connection that will support the 3ds. Just set it up and connect.