That thing is in really good condition! I would open the top case and have a look around and see maybe it's a loose connector or something. Also check for corrosion or liquid damage. Have you tried an smc and pram reset?
I can't remember off the top of my head but when you do an smc doesn't the clock and date reset? Or is that a pram reset?
Also I heard someone day with the batter as charger connect if you hold the power button down until you hear a loud beep thats also some kind of reset but they may have no clue what they are talking about n
I would replace the DC board an also the battery connector to start.
http://www.ifixit.com/Guide/Repair/Installing-MacBook-Core-2-Duo-Battery-Connector/525/1
Darn that's too bad.... Sounds like logic board to me. Hmm how are the solder joints where the battery connecter connects with on the logic board?
If there's water/moisture in the port while charging, then it would be an issue.Is charging corrosion a common problem? Have experienced this in a lightning port and was told by the Genius Bar rep it's due to water/moisture in the port while charging...