Audi designers used the 8J spoiler as a sought of non electronic telltale (excuse the pun)
To quote Pg 39 of the 2008 TT Training Manual Electrical system and Comfort System Electronics
System Malfunctions
"The rear spoiler is extended if any one of the following faults occur:
-Comfort System Control Module J393 does not receive a road speed signal
-J393 is unable to receive any CAN messages.
- Implausible signals received from switches F135 and F136
- Less than 10.5 Volts at J393"
Switches F135 and F136 are pins of the centre console rear spoiler switch
So it appears that your Comfort System Control Module is the most likely culprit.
Having said all that the rest of the symptoms are consistent with a flat battery in a TTS