A friend of mine moved her xbox to another room and lost all of her DLCs. She only gained internet access to download them and does not belong to xbox live. Does Bethesda keep track of who has paid for a download and can she re-load them? Anyone else have this problem?

It shouldn't dissappear by simply moving the console. Did you accidently remove the harddrive? Try removing in fully and putting it back in place.

Beth does not track what DLC your friend purchased, but Xbox does. You don't need a gold account to download DLC but you do need a basic free Xbox Live account, so she must have had one at some point. If she logs into her Xbox live account, she should be able to re-download the DLC for free to the same Xbox, since she already purchased it. It sounds really strange however that the DLC would disappear just from moving the console to a different room.

Perhaps she never finished the installation of the DLCs, they may need to be activated for the on the first launch.