Hmmm... interesting. A couple/few broken strands could increase resistance (which I'm guessing what the sensor measures) which could screw up the reading... I've also heard when the batt temp sensor goes bad that it causes strange electrical problems. Will have to check that out too.
I don't believe that is the battery temp sensor....but I have a CRD so I could be wrong. Battery temp sensor is two wires only...I will look up the colors of the wires for you sometime.
This is not important....firstly if there was a problem with the temp sensor then the red battery light would show up on the cluster...the same light as when you have a charging problem.
Secondly the battery temp sensor is there purely to slightly adjust the rate of charge from the alternator to the battery depending on the temperature....it is not a big deal and has nothing to do with your problem and does not cause weird electrical problems.
Can you tell me exactly what color wires are shown in your picture ie. White with Red stripe etc.?
You need to find out why you have the check engine light on...you need to beg steal or borrow a scanner to see what your Jeep is actually complaining about and your problem with the wipers is probably something totally unrelated.