Display didn't work before pulling battery now it works fine?

I bought a Macbook A1181 locally through craigslist. Actually I am currently using it to type this forum thread but anyways back to topic. I bought it with two major problems. Display didn't work and the battery wouldn't hold a charge. The Macbook will only power on when it is plugged into the AC charger, if I try to power on the Macbook with the battery and the AC charger unplugged it won't do anything.

The AC charger has a green light whether the battery is installed or not. Wierd thing was that the display wasn't working before I pulled the battery, it was extremely dim to the point that you could barely see what was on the screen and at a angle you would finally be able to see that the display still worked but the backlight/inverter was faulty, at first glance it appears that the display is blacked out and not working at all.. Well I decided to pull the battery to start diagnosing the battery problems, booted it up... Display starts working? It has been nearly 24 hours and I haven't run into any issues with the backlight/inverter/lcd, it seems as if nothing was ever wrong with it. Does anyone know what the problem is? I am so confused that I don't know why it would randomly start working after the battery was pulled?

Part two of the question is, since the laptop will only power on when plugged into the AC charger this is an indicator of a dead/failed battery that won't keep a charge. It isn't even being detected by the system.

I hope I was descriptive enough to receive some reasonable feedback on what I am experiencing. Hope we can all figure this out!