Did a couple more tests:
- battery at 11.6V with the engine off, 13.9V with the engine running. Doesn't vary much with load or rpm
- with key in on position before starting the engine, the brake, ABS, SRS and check engine lights come on. The ABS and SRS go off after a few seconds. The battery light glows dimly while cranking. The high beam, brake pads, oil level, coolant level and bulb lights don't come on with the key in this position.
Also, the radio keeps coming on on its own. Every time I start the car and even after I turn it off with the engine running. I mean the engine's running, the radio comes on, I turn it off, it comes on again... and again... and again.