04 Jetta 1.8t Electrical Issues

Hi, I'm doing this for my sister, her car is giving her problems.

She had her codes read, at autozone by an employee. She wrote them down and looked them up. They related to the CTS, it said something along the lines of Coolant Temp Sensor- Input High. She bought new sensors and replaced the one by the back of the engine, with 4 prongs. She still has the 2 prong sensor waiting to go in, I haven't taken the time to find where its located.

Two days ago her battery was dead. We took it out, trickle charged it and put it back last night. Car started fine and ran properly. This morning it was nearly dead again, just barely able to turn the engine over, and not fast enough to start it. After she pulled the key out of the ignition I noticed a slight humming sound. It was coming from what appears to be a pump, attached to the radiator. It definitely connects to coolant lines, and I can only assume its a pump... dont know what else it would be. Has anyone else had this problem before? It ran for at least 5 minutes, and was still running when I shut the hood. I've since disconnected the battery. Could this be related to a possibly faulty CTS that hasnt been replaced yet? Does the fact that it was running after the key was removed mean there are bigger problems? Also, does anyone know where the second CTS is located?

If you need more info, please ask. My sister has to drive to work everyday... and I've become her chauffeur. I would like to see this fixed as much as her.
Thanks.