Unusual power and LCD backlight behavior with MacBook Core Duo

It was one of the first ones, with 2x256 MB and a Combo drive.

When I plug in a known working AC adapter, the LED on the Magsafe connector behaves as if an inserted battery needs charging - it goes from green to amber. Problem is there's no battery inserted.

I power it up and hold down the power button to reset the PRAM. Upon chiming, the backlight will flicker once. OS X loads up with the Apple logo and a rotating gear icon without a backlight. When the blue background appears, the backlight activates but only momentarily. As the backlight goes out, apparently the MacBook goes to sleep. The sleep light's on, the Caps Lock LED is deactivated, and there's no image. Pressing Caps Lock will activate the green LED, but it will go out again.

With a known working battery, I can't get any reaction. This MacBook will not boot from a battery and will not charge a battery (the pips never activate upon insertion).

I'm focusing on the Magsafe board AND the battery connector. I'd love to hear other thoughts, too.

Thanks for reading!