x1 is fine for mining. PCI-extenders just use a ribbon cable to allow you to raise your card from the board.
Application-specific integrated circuits, or ASICS, would be much more energy efficient for mining. As far as I know no one has done this yet. Field programmable gated arrays, or FPGAs, have been used. These also are highly efficient but can be prohibitively expensive... though cheaper than an ASIC.
Are you sure 1x is OK for mining? No card seems like it'd fit that slot. Do you mean 4x?
The card wont fit into the x1 extender on it's own, the end has to be sawed off.
Some sellers already do it for you before shipping though.
Or I'd be better off using the 4x because that and the 16x are doubles paced anyway. Right?