I have the same exact problem with my iPhone 3G. It also thinks headphones are plugged in most of the time, so I'm thinking perhaps it keeps detecting headphones, which then wakes up the screen....does yours also have that problem? I haven't found a fix yet though.
Yes, LED christmas light strands are often two independent circuits. The LEDs are connected in series, so if one is disconnected or somehow failed open circuit, that circuit will go out. Of course the same will happen if there is a break somewhere in the wire. You could check with a voltmeter to see if you are getting 120vAC out of the socket on the tail end, which if zero would mean a wiring break. But I think chances are you have one failed LED. Further testing would mean cutting into the strand.