The charger should not overheat from being plugged in for too long. The only reason that a battery charger would overheat if plugged in for a long period of time is if it was malfunctioning.
The battery charger's output Voltage can be tested with a multimeter set to DC Volts to see if it is working or not. Here is the polarity of the TaoTao ATE-501 scooter's battery charger output plug.
The TaoTao ATE-501 battery charger should have around 56 Volts at its output plug when it is plugged into the wall. If it has zero Volts then it is faulty.
If the battery charger has the correct output Voltage then the next step I would take is to physically inspect the batteries in the battery pack to make sure that they have not swollen or melted. If they have then they should not be used any longer and be replaced.
If the battery charger has the correct output Voltage then after inspecting the batteries and if they look good the next step I would take is to perform a more comprehensive test on the charger and battery pack. To perform this test the charger should be plugged into the scooter and then plugged into the wall. Then the Voltage of the battery pack can be tested with a multimeter to see if it slowly increases while the battery pack is being charged. You will need access to the battery pack terminals in order to perform this test. If the battery pack Voltage does not slowly increase while it is being charged then the charger could be malfunctioning or the batteries could not be accepting a charge due old age or internal damage.
If the battery charger simply failed and stopped working then the batteries may not have received a full charge which is causing them to be half charged. However if the battery charger is working and the batteries are only half way charged then that could be caused by a faulty battery charger port or faulty wiring between the charger port and the controller, or it could be from the batteries being old or worn out. The comprehensive battery charger test described above will help to determine if power from the charger is reaching the batteries and if they are accepting a charge.
The batteries could also be load tested to see if they are in good shape or worn out. Here is a link to our electric scooter battery load testing guide: http://www.electricscooterparts.com/loadtest
Please let us know if you have any questions.
Great to hear that you finally got the batteries out of the frame. Sometimes when the batteries are swollen removing them from the scooter can be virtually impossible.
TaoTao electric scooters are built on the same platform as most other Chinese-made electric scooters are though, so the process of removing the rear wheel would be almost identical to that of any other similar street legal type of electric scooter.
I found a video showing how to remove the rear wheel of a GIO electric scooter which should provide most of the information that you are looking for. Here it is.
Please let us know if you have any questions.
Aaron verity