No it does not have to but it it is recommended!
N it doesn't use Internet
Yes, as long as you have Internet connection though. If you goth GameStop you can buy minecraft for the Xbox. They will give you a code, you enter this in the Xbox menu where it says redeem code.
Yes, you can have up to four players on one Xbox, even with no internet connection.
Yes. The seed option is tied to the game, and not the internet.
You can only install items without the internet, but downloading items and updates needs the internet as well.
Yes, there is split screen on Minecraft Xbox 360 Edition, but it is exclusively for the use of HDTVs, so far.
No, you have to use your gamertag.
there are no mods for minecraft on xbox
You could get a copy from the internet and put it on a USB stick and run it from there
No.
NO you can also use wired
Connect your xbox to the internet, sign in to your xbox live account. Launch the game and if your using an old version the game will automatically update. You will be prompted to confirm the update.