Q1: My gas gauge stopped working. How do I tell if it is the sensor in the tank, or something in the gauge?
A quick check is to remove the green wire from the sensor at the tank. The VDO instrument should then read full scale if the instrument is working and the wires are correctly landed. A further check is to ground the wire (green) at the tank , and the instrument will read nada. Blu/yel on +, grn on G for gauge on the Combination Instrument.
A check of the sensor by an ohmmeter will record 200 ohms with a full tank, and 6 ohms with an empty tank. There can be some variations of these numbers. Still they will be close. So now you can test the instrument (gauge) and the sensor.
If the sensor doesn't read correctly, it's likely that the wires inside the sensor are to blame. They are broken, shorted together, or the pivot is dragging. I suggest you replace the sending unit, trying to repair is frustration.
Q2: My gas gauge jumps around, but the gauge only jumps when the tank is less than completely full. I top off the tank, and the needle is pegged fully to the right (above full).
If when the sender float is pushed to the top of its limit, and can't move, the gauge stops jumping, logic would suggest that it is the sender that is bad.
One 356er, when refurbishing the outside of the fuel tank, left the sender in it, and was turning it upside down, on its side, etc. That banged the sensor around a bit too much and it began to exhibit the symptoms. A word to the wise - take out your sender before you start grappling with the fuel tank.