It says the cube is immoble. If a character rolls the relic roll and succeeds can they move around with the cube?

I believe the cube is put on their character card, so technically it moves with the character as far as that goes, just like Globe of Ultimate Knowledge and Power Gems.

Technically, it doesn't move because it's not on the board. It isn't being carried, it just sit on the card, which is not on the board in any way. (You did say "technically". : - )

Its in Howard the Duck's pocket.