2000 Oldsmobile Alero Good battery but no system electrical

There are three issues which may be related or may just be coincidences. I have noticed recently that I receive a static shock after exiting the car and touching the door, etc., as if there is a grounding problem. Then about a week ago the car started some peculiar behavior. Occasionally after starting (and beginning to back up) the anti-lock warning would come on and stay on, though the brakes worked normally without any indication of AL actually occurring. If I turned the car off at this point, it would not start without a great deal of manually goosing the gas pedal, as if the fuel injection system was out of whack. Once it started, everything was ok.

Then yesterday I go out to start it and...nothing. And I mean no electrical anywhere; no acc, no power windows, no auto lock/unlock of the doors, no lights, etc. Completely and utterly dead. Thinking the battery (which is new) may have somehow drained, I jumped it, and it started. I drove it for several miles, and the battery charged and even started the vehicle normally after shutting it off. Then later that day I go out to start it and...nothing again. This time not even a jump had any effect whatsoever. I checked the battery terminals with a multimeter and I can read 12v (it actually says 13v??), so I'm assuming the battery is ok.

This has me really stumped. Today I'm going to clean the battery cables and check fuses and connections to the ignition switch if I can find it, but otherwise, I'm in over my head. Ideas?