Assuming that Audis have more than a passing similarity to their VW relatives, there will be a coolant level sensor in the coolant reservoir "ball." The sensor is a pair of probes that look to be made of stainless steel that come down into a small coolant dam in the tank. You can easily see if they are there by looking down from the top when the tank is open.
If they are there it would be worth cleaning them with a bit of ScotchBrite or something similar. They sense the presence of coolant by detecting (or maybe even measuring) conductivity between the two probes. If they become contaminated they can fail to work.
So, if the coolant level looks good, I'd start by cleaning them. It's quicker and less expensive than replacing them (which I think means replacing the coolant reservoir).
The temperature glitches suggest a few things to me:
- Damaged wiring between the sensor and the gauge, or
- Maybe air in the cooling system – I'm not sure exactly what to expect but if there was air trapped in the system and it caused the temperature sensor to loose contact with the coolant it seems that you could see the reading jump one way or the other. What I'm not sure about is if the pressure in the cooling system is high enough for the temperature to get all the way up to 120º C.
- A failing sensor.
Since you've swapped the sensor it seems that one of the first two is the most likely explanation. I suppose it would also be possible for air trapped in the cooling system to cause enough disruption in normal liquid levels that it might cause the coolant level sensor to give you false alarms.