Crank no start?

Okay, but all the pins your testing are input pins we need to know power is coming out of the ignition switch, Pins 2, 3, 6, 9, 13, 14. Pin 10 is output to the power windows, Pin 5 is the Key in the Ignition switch sense circuit. There has to be power coming through the ignition switch on all the Output labelled pins. Pin 13 will only have power while cranking. All those output feeds go to certain fuses in the fuse/relay panels. This is where pins 2 and 13 go from the Ignition switch.
That yellow connector might be for the airbag, so I wouldn't mess with it too much until I can figure out what they are for.

They are for the airbag, do not touch that yellow connector or you might trigger the airbag and it will go off in your face. That's a serious hazard.

Okay, 6 is a ground, so use that for your ground location while testing to verify its good. Or hook to a 12v source and touch pin 6, you should read 12v. The Key in Ignition Switch sense might be the issue. If the PCM or security system does not sense the correct key or any key at all, then its preventing starting from the Ignition switch. I'll check what is supposed to be there for a voltage signal. I would think just 12volts. But ill verify that.
Ok the way they word this is a little confusing, But the key in sense closes a path to ground for whatever this Central Timer Module is, so it looks like pin 5 should create a Ground when the key is inserted, the service info here states something about the driver's door being open, but it doesn't really make a lot of sense as to what they are doing with the door switch. For testing purposes, I would try to tape the driver door switch closed so it's not messing with testing the Ignition switch. With the way that was overheated. I would be surprised if it's not damaged. But try getting the door switch to stay closed while checking that. I'm going to dig around for more info on the operation of this circuit.
Okay, the 3rd diagram shows what pin 5 and 6 do, they complete a Ground to the BCM, so with the key in the Ignition switch there should be continuity from 5 to 6, and 6 to ground.
If all that checks out ok, then we need to go directly to the PCM connector C3 pin 3 and see if the PCM is giving the ASD relay ground. If it is, then there is an open circuit between the PCM and the ASD relay Ground for pin 86.
PCM connector C3 should also read an output from the ASD relay on pins 19 and 28, they should read the 12volts coming out of the relay. You can see this in diagrams 4, 5, 6.
So, I would back probe C3 pin 3, it's a brown/white wire and it should read 12volts until you turn the key on and then crank it should go to 0volts because it's activating the relay and being pulled to ground.

Let's go right to the PCM and check there, because you did have a code for the power relay being an open circuit. Let's make sure this is not an open wire somewhere causing all this.

No, it means one of the wires coming from the ASD relay is an open circuit. Thats why I posted the PCM C3 connector diagrams in my last post. What other codes did you get?
If you back probe the C3 pin 3 wire, that is the wire that goes to the ASD relay pin 86. So it should read 0 volts when the relay is activated.

Okay, give me a couple minutes I'm going to make a diagram to show exactly how to do this, but you will need to remove the back cover off the PCM connector C3 so you have access to the wires while the connector is still plugged into the PCM.

It's okay, when you're looking at the PCM, it has 4 connectors, In the 2nd diagram you're looking at the back side the connectors plugged in, when you unplug the connector from the PCM you'll see that is has a plastic cover on it, those are just clipped on, and to access the wires going into the connector you need to remove the cover, then you'll see all the wires running into it and you can plug the connector back in, and I use a T pin to slide down the wire into the connector, making sure you don't touch the T pin to anything. But that's how you get a reading at the PCM without having to pierce the wire.
The 3rd diagram I made a while ago shows how to back probe a MAP sensor connector, same idea here, except you need to remove the back PCM connector cover, Sorry for my terrible artwork. But does that make any sense? If it's easier for you, we can just try a resistance check from the PCM connector disconnected (pin 3) to the ASD relay pin 86 and see what the resistance of the circuit is.

You mean you pulled the ASD relay out and tested the pin locations where the relay sits? 85 and 86 and got 123 Ohms?

Okay, I was going to have you unplug the PCM connector and test from there to the ASD relay pin 86. So, you're checking just the resistance from ASD pin 86 to PCM connector C3 pin 2(diagram 4) shows its location. Make sure the key is off before disconnecting the PCM connector. Also just touch your meter to the PCM connector pin, don't jam the meter lead into it or you will spread the connector pins and they will never make contact again. So, check on the Ohms/Resistance setting not voltage. I'll get back to you shortly, I need to run out on a mobile call real quick.