Don’t know if you’ve received any help yet. I have a 2004 Audi A4 B6 1.8T. I had a bad temp indication and it was my temp switch located on the thermostat housing assy at the back, between engine and firewall. Took some work to get access to it, but wasn’t very expensive. Make sure you get a new o-ring with the temp switch (sensor).
It appears you are holding the sender unit for the outside temp indication. The post above I think is referencing the coolant temp. As a test for the outside temp problem, you might try bridging the two plug the terminals and see if you get a reading on the dash. If you do, the prob is either the sender or the wiring. Good luck.
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