Is it possible to measure Spinal Cord electrical activity over skin?
Long story short: No, at least not with today's tech.

The signal to noise ratio is going to be very low, and there is no particular geometrical mapping of activity that you can take advantage of for surface recording.

There is a bunch of research, and mini consumer products, that takes advantage of the geometrical mappings in the brain to measure EEG to control cursors, robots, and such, with different levels of success.  

There is also a possibility of grabbing the spinal cord signal at the ventral root, as it leaves the spinal cord, and thus much of the ongoing "noise" of the spinal cord is filtered out, and you're left with what's exiting at that level.  I haven't heard of this being done with surface electrodes (or even implanted electrodes, for that matter).

Impossible??  Well, many things are possible.  The way to go about doing what you envision??  Probably not.