Hello & Welcome to JA,
Ok, the first 2 codes are evap codes & will not cause a driveability problem. Th e second 2 are the problem -DTC U0107 Lost Communication With Throttle Actuator Control Module These 2 faults are closely connected. To find this problem one will need to view the tps ,possibly when this event happens. The repair is usually replacing the throttle body., if all other data is good.
P2135 - loss of throttle actuator control data
The throttle body assembly contains the following components:
2005 Pontiac G6 6 Cylinders 8 3.5L SFI.
To answer your question about why stuffing a rag in the gas tank causes the problem to go away, I can only say.
The 2 codes P0443 and P0455 refered to the Evaporative Emission (EVAP) Purge Solenoid Control Circuit and Evaporative Emission (EVAP) System Large Leak Diagnostic. Perhaps the original cap is bad, have you tried a new gas cap. The EVAP is a closed system and without the cap or a faulty or leaking cap, the system would loose pressure or vacuum. Plus, when you stuffed a rag in there you must have been making a good enough seal.
After you forgot to put the gas cap on did you ever clear the code? Did the cap ever fall and hit the ground? As far as the engine loosing power, you may have more problems then a code will show. I would start off with replacing the fuel filter, clearing codes and checking the fuel pump pressure.
I wasn't even looking at those other codes cause it was already answered. I was only trying to answer the one question that you asked after relisting. If you want I'll look into it.
Ok, its back up. I see the "Reduced Engine Power" found under the code P2135 and U0107. Reduced Engine Power: The scan tool displays Active or Inactive. The scan tool displays Active when the PCM receives a signal from the TAC module that a throttle actuator control system fault is occurring. The PCM limits the engine power. For more info, click here. For P2135, Possible Causes: The throttle position indicated by TP sensor 1 disagrees with the throttle position indicated by TP sensor 2 by more than 6 percent. OR The PCM learned minimum throttle position of TP sensor 1 disagrees with the learned minimum throttle position of TP sensor 2. OR The PCM detects that the TP sensor 1 signal circuit is shorted to the TP sensor 2 signal circuit.
For U0107, Possible Causes: Invalid or missing serial data messages are detected for a predetermined period of time.ORThe throttle limit is exceeded while operating in the Reduced Engine Power mode.