Sounds like a bad battery, but if it stays green when not plugged into the laptop the magsafe is faulty and potentially dangerous. Do not continue using it, get it replaced under warranty or buy a new one.
Yes. The light on the charger is constantly green when its plugged into an outlet whether it is plugged into the computer also or not. I looked through the forums and didn't see anything like this yet.
Yes. The light on the charger is constantly green when its plugged into an outlet whether it is plugged into the computer also or not. I looked through the forums and didn't see anything like this yet.