I am NOT a mechanic.... but do have a basic
I think you have a pretty good handle on what is going on. However, I am afraid I have some bad news for you. The problem with the "Transmission Control Relay" is a simple matter of "symantics". Your vehicle IS equipped with a transmission control relay, but the factory information refers to it as the "EATX Shutdown Relay". Same thing - different name - and dealers are not real smart when it comes to these things. Dealers usually do not see the problems that we see in the "real world" with high mileage cars. Anyway, the EATX relay is located in the Power Distribution Center in the engine compartment.
Then you have the wrong information about codes 36/52/54. (you actually said "19, 36, 52, 54" then said "36/54/42" - I am assuming that the "42" was a typo and the "52" is what you mean.)
The definition of code 36 is "Fault Right After Shift". This code NEVER sets alone. It is ALWAYS accompanied by at least one "speed error" code (codes 50 - 58) As you can see, this is holding true in your case with codes 52 and 54.
Code 52 is "Speed Error In Second Gear"
Code 54 is "Speed Error In Fourth Gear"
These codes can ONLY be caused by internal transmission problems or a faulty Transmission Control Module. (or something the troubleshooting charts do not account for...read on...)
Please not that these are NOT speed SENSOR errors. If there was a problem with the speed sensor it would set codes 56, 57, or 58. These codes can only set when there is a good signal from both the input and output speed sensors. The TCM compares input speed along with the commanded gear ratio, to the output speed. The redings should fall within certain parameters for each gear. If they fall outside the designated window, it can only be because something is not engaging or is slipping inside the transaxle.
On the other hand, I have seen similar problems caused by low-voltage/voltage-drop problems - which brings us back to the relay thing and the communications buss. I have seen bad ignition switches and/or burnt connectors at the ignition switch cause this type of problem. When I checked for voltage at the transmission connector, I had the correct voltage. But when the system was loaded, the voltage dropped drasticly. Come to find out, there was a bad connection internally in the ignition switch which would allow battery voltage only as long as I was checking it with a digital multimeter. When I loaded the circuit with an incandescent test light, the light was a little dim as compared to how bright it was when I touched it to the battery post. So I loaded the circuit with the test light and performed a voltage drop test on the circuit and found that I had a 3 volt drop. Replacing the ignition switch fixed the car.
So, in wrapping this thing up, before tearing into your transmission, check all of your input voltages to both the PCM and the TCM. It is best to disconnect the connectors and load the power circuits with a test light and perform voltage drop tests accoss all of them. Don't forget to also check the resistance of the EATX Shutdown Relay contacts as well - or just put a new relay in to see if it makes a difference.
The good news is that I believe you will have the thing fixed when you correct the communication fault - I'm TOTALLY with you on that one. The book just doesn't bother to mention that voltage drops can cause communications faults. (Computers do really weird things when they are not fed properly.)