> When I turn it on, it comes up with a pattern of horizontal stripes, 
> about 4 pixels thick, for the entire display area. It doesn't do anything 
> else after that.

This is very interesting .. my MAC 512K started doing the exact samething.

Did you try replacing your battery?  I don't know if it would make any
difference, but my Korg POLYSIX synthesizer's batter died and the synth
went totally wacko..  (an analogue).  I replaced the battery and it
started acting sanely again.