A lot of the 04 engines were subject to a recall to replace the fragile ceramic glow plugs with more robust steel plugs. The steel plugs were 5 volt as opposed to the original 7 volt ceramic. There was a reflash of the ECU to adjust the voltage, but that often resulted in hard or no start when cold.
VW eventually came up with a newer version of software to make the engine start when cold, and at the same time replaced the 5 volt steel plugs with 7 volt steel/ceramic ones.
It may be helpful for you to contact a dealer and find out which, if any, of the glow plug revisions have been installed on your car.
Another possibility is the PO replaced the glow plugs with the 11 volt plugs as used in ALH engines.
The temperature sensor has two sections, one for the gauge and one for the ECU. To test the ECU portion, you will need to have VCDS and compare the coolant, fuel and intake air temperatures after the car has been sitting overnight. If one is much different from the others it is suspect.
If there was a bad plug, you would have a CEL, so that is not a problem.