If the alternator was replaced before the leaks, it's possible that ruined it. However, before you go down that road, take it to AutoZone, they should have a charging system tester that can tell you if the alternator is good or not. Before you go, read though this for background:
http://www.lextreme.com/alternator.html
I also have the Duralast Gold in mine and am happy with how Duralast batteries perform.
If the car had problems before it sat, who knows, but if it was "given" to you as you say, it's hard to complain. Well, not too much anyway.
However, I would still investigate the trunk wire harness issue to see if you don't have a bad wire in there. It doesn't necessarily have to be the white one though.