I think if the OP reads through the TSB Lucky, the solution is in there, but at this point my recommendation is that the HHR be taken to a dealer or independent GM Tech.

The cascading nature of the reported faults leads me to the same conclusion as Oldblue, some quality time between the vehicle and a Tech II is required.

At the very least an experimental "Lobotomy" with the NEGATIVE battery cable disconnected for thirty minutes to return every module to the preset default values may be worth trying.

 

I think your right