Your display name is essentially your username that is shown to all players above your players head in online multiplayer.
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Unfortunately, you can not change your name on Minecraft!
It doesn't mean anything specific 'in Minecraft', it is short for Major League Gaming.
What do you mean by this? If you simply mean have minecraft on more then one computer then of course you can. As long as you have a valid account you can download minecraft onto any computer.
If you mean the box you purchase it in, there is no box, you can only buy it from minecraft.net, and if you mean crafting a box in minecraft, it can be as big as you want, there is no determined size of a box in the minecraft community
Clear all the Minecraft folders on your computer and don't use the same Minecraft name :D