I have the same/similar issue with my '06 A3 8p. However, with mine the blower stops (dead air) for no reason, and then later comes back on. Wehn that happens the climate control fan speed won't respond, showing the first tiny bar on the fan speed scale. Pressing "+" to increase speed does nothing. Pressing "-" to reduce fan speed turns off the climate panel. Other buttons seem to work, but it doesn't make the fan come on, whether I have it on Auto, Econ, Vents or any combo - still no airflow. I've read about fixing/replacing the blower and module, but most talk in terms of their climate controls still working, i.e., cranking up fan speed, the display showing a faster speed, just nothing happening at the fan. Mine doesn't do anything on the fan speed display when cranking up fan speed.
I'm wondering if the fan control module provides feedback and therefore the speed display is actual speed, meaning if the fan is stuck (maybe debris in there) the display won't show the higher speed requested?
Any advice/suggestions are very welcome before I start tearing my dash apart.