No, you have to use your gamertag.
No, the Xbox version of Minecraft is for Xbox360
There is only one version of Minecraft for the Xbox 360.
The PC version of Minecraft can be downloaded from Mojang's official Minecraft page. The XBOX 360 version of Minecraft can be downloaded from XBox Live Arcade.
Currently, the Xbox 360 version of Minecraft is running 1.6.6 beta. This update came out before the adventure update of 1.8. It will take a while for the Xbox 360 version of Minecraft to catch up to the PC version of the game.
Yes there is.
No, the xbox version is 1.5 patch.
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.
You can't. This is because the version of the Minecraft Xbox 360 is different from the PC Version.
You can't mod the XBox 360 version of Minecraft.
Yes it's a completely different system and your Minecraft on Xbox will be connected with your XBL account.
No. You can't mod the XBox 360 version.
Minecraft Xbox 360 edition and Minecraft (PC) are two separate games; if you want the Xbox edition, you'll have to pay with Microsoft Points. If you want the PC version, you'll have to pay for a Mojang account. It's best to be aware that buying any one version will not grant you license to download/play another version; you have to buy each version separately, the same with the mobile/tablet version and so on. There isn't a Minecraft Xbox 360 disk; you buy it with Microsoft Points via the Xbox Live Marketplace: only an internet connection is needed. Not sure about whether or not it requires a Gold membership as I only have the PC version and have no plans to buy any other edition.