A new case is for additional mounting options, essentially you would get a case that has more PCI expansion slots than the motherboard. That way you can mount a card in the empty slots below the board, leaving room for the hard socket of the extension. You can certainly get that ribbon cable underneath another card to use the available 1x slot.
Hard to say without knowing the case or the motherboard. Assuming an OEM computer that could be an ITX board or one of those odd inbetween ones with two slots ITX+1 or MicroATX -2. If not, same applies, really. Need room for an expansion card, but the slot is physically occupied by the GPU. Get more slots.
If the computer is actually BTX, then it should just die in a fire and be replaced...