Just a few guesses / comments:
- Having the AC in auto while heating: it is normal that the AC compressor turns on. It is not "fighting" the heating per se, it is dehumidifying.
- Turning off the AC and then turning down temp control: it is normal that the AC does not turn on. This is the "ECO" mode, cold setting simply means no hot coolant diverted into heater radiator.
- Setting the temp to 20C produces a steady stream of cold air, does not matter if gets too cold in the cabin: not normal. Could be faulty internal temp sensor or a computer glitch (set temps in F?!)
Ps: Please note that I do not own a Superb MK3 (yet...) Been through a bunch of cars though. Answered the original post because no one else did and wanted to keep hope alive for a more educated comment