Here is another look at the wiring; the red goes to the starter and battery the purple is the solenoid signal.

https://cimg3.ibsrv.net/gimg/www.che...c155f04daf.png

I didn't make it over to where the HHR is today to look at the ignition wiring. I am going there in the morning to inspect. Another update I have for the situation; I spoke with the guy I purchased it from and he claims that it had no electrical issues when it was still running. Says the alarm was removed when he bought it (6 years ago) and no adverse effects from its removal.He did offer that there was an incident after I paid for it, but before I took possession.Since the battery was dead, he was unable to get it out of park to make accessible for tow truck. He removed the trim ring around shift lever to access the override for the lockout. Says harder than remembered it being in the past (evidently happened before) and had to use more force. I know the neutral safety switch is on top of the transmission, but is there voltage at the shifter that could have been damaged into a signal wire causing the "back-feeding" I have at the RUN/CRANK relay terminal?

Not that I know of. But there is an extended warranty on the shifter sticking issue.
From your description there is also something energizing the starter relay also. See the diagram posted today.

I will determine which port is being given the 10.4 volts and follow it. On first diagram you posted (if memory serves, I am not near the HHR) it the E2/C4 location. So if I remove the 15A ECM/trans fuse and it still has voltage, I know the issue is after the box and if not then it is before. That sound correct? If it is the F2/C4 location I do not see a fuse in line on diagram to help narrow search. The reason I offer these 2 locations is because I am certain it was next to the constant voltage location and not diagonal from it. Thanks again to the replies and PMs.

I looked at the HHR today and verified that removing the TCM/ECM fuse stops the "back-feeding" to the relay. I searched the wiring loom as best I could and see no breaks or damage. I am curious if the transmission control model could be the culprit. I see that someone has attached a ground to its case. The eyelet, at least, is not factory. The ground wire itself disappears into the loom. From the above diagrams I think it should be grounded at strut tower. I see no evidence that there was ever any ground at the tower.

My local Pic a Part has just put 2 HHRs out in the yard. I plan on traveling there and comparing the wiring. If any feels the TCM could be at fault, I can get one and swap it to test.

That grounding proves that the previous owner lied to you; there have been electrical problems for a while. That is a common youtube bandaid that is useless.

Are you saying there is voltage on the ground side of a relay?

Check the connectors to the TCM and ECM for oil or water or bent/loose pins. One member found that oil was somehow migrating up the loom from the ground on top of the trans.

From what I think you are saying the common point is the park neutral switch.

Again pardon my ignorance in describing the locations. But I have 10.4 volts at the location on the 1st diagram above marked ECM/trans fuse. If I am reading the relay correctly, that should only energize when commanded, right?

And as for the ground issue, I should remove the ground to yje TCM case and wire it to rop of trans on a clean location?

With the ground removed from the case, communication is lost to TCM. When I return tomorrow I will inspect the ground at transmission. I will also inspect the neutral safety switch. I assume the issue there would be internal and not apparent.

I think that first diagram is wrong, the 2 don't seem to agree. I don't know where 10.4 volts might come from; 12V, 5V ref and the radio controls use 10 V ref.

Maybe the wires got switched on the park/neutral switch?

https://cimg9.ibsrv.net/gimg/www.che...1979ce51df.png

Spent some time at the HHR today. I picked up a used neutral safety switch from local pull it yard and installed it on the HHR. It cured the back feeding issue. All grounds seem in perfect condition, It will no longer crank though. I had opened the steering column area up to access ignition/starter switch and found that the connector to switch has been crudely spliced in. If I hold the harness in a certain position I get power to dash in the form of flashing turn signal or hazards, brake fluid on info screen, solid security icon, etc.

I obviously need to properly replace the pigtail and re-inspect. But I am frustrated by these new issues after the joy I felt after curing the back feeding issue. Now I cannot start the vehicle. before I couldn't turn it off. HAHA.

Does replacing the neutral safety switch OR ignition/starter switch require a security relearn procedure?

No, but the PNP switch needs to be correctly aligned, there is even a Kent-Moore tool for it.

https://cimg0.ibsrv.net/gimg/www.che...ed055d5b80.png
There is a procedure for removing the ign switch.