Maybe this will help:
http://www.veryuseful.com/mustang/te...Radio_diag.gif
The lack of memory sounds as though who ever hooked up the stereo did not run the constant power wire correctly. It appears as though they put it on a circuit that is activated by the ignition.
Is it aftermarket, or the stock stereo? If aftermarket, I would re-wire the entire thing properly. Hook the yellow wire to a constant 12V source, the red to a ignition source, the black to a ground, and then just wire the speakers accordingly. If it is stock, I would replace it with an aftermarket head unit. If you are just looking for some tunes, a decent cd/mp3 player can be bought for cheap.