When the ignition is turned to the on position and the dash lights come on temporarily the check engine light does not come on?

No, I just wanted you to do a visual inspection of that ECM connector, checking to see if any of pins in the connector were pushed into the connector. Or if the ECM pins had corrosion on them or were bent. If someone else has been messing around with it before you bought the vehicle, they could have done anything. We haven't gotten to any voltage testing yet, because just unplugging that coil yesterday sent the vehicle into a no start condition, tells me that you leaned on, or pushed on a connector that caused that. Just the same as you not having a check engine light.
I just put the diagrams up to show how the circuit works and why you need to check the ECM C1 connector for pin fitment issues first.
Same with the Fuse box, does it have loose or corroded connectors?

Okay, I'll take a look for the diagram of the fuse panel, as for that test light I can't tell if it is an LED or not, I don't think it is, If it has a regular little light bulb in it, then it's not a low current test light and you have to very careful testing computer driver circuits and pins, you can burn out a driver easily. You don't want to do that. I did some reading up on the MIL and the Cluster supplies voltage to the LED in the instrument panel and then the ECM grounds the other end to turn it on, so either the instrument cluster is not supplying that voltage, or there is a problem in the circuit further down. But there is a testing procedure, we will have to modify it somewhat because I don't think you have a scan too that can turn the MIL with a bidirectional command.

Diagram 1 is the Fuse block, I marked as many relays as I could find, there is a pinout list for each connector that's underneath, but not layout of the top of the fuse block.
Diagram 2 is the underside, with 6 main connectors, these connectors tend to get corroded because of location, water and stuff from the road splashes up, and that's what happens. So, if the fuses up top look dirty, imagine what those connectors look like. If you can unbolt the Fuse block and take a look at those connectors, you might find something.
Pins and wires break under fuse panels like this all the time. It causes relays to lose power or control. I have seen this happen many times. But pull out the relay that is just sitting there and see if there are any pins below, if not, then you know that someone else has been trying to fix this vehicle in the past and has had some type of electrical issues.
We can check the ECM connector tomorrow, or whenever, with it plugged in and back probe the pin 68 and see if the ECM pulls that circuit to Ground at just Key On to begin with. I'll explain how to do that with the multimeter, we will just check for voltage drop to the battery negative with the meter.

The connector C109 runs from the ECM under the Fuse Block to the Instrument Cluster, there's 2 different designs of this connector (diagram 3 and 5) but the pin A7 (Brown/White wire) is the same, this is the MIL command wire that gets grounded by the ECM to turn the MIL On. This connector should be checked as well. Its black, the 1st design has 38 wires, and the 2nd design has just 24. But it shouldn't be too difficult to find.

Are there any pins if you pull that relay out? Or is it just sitting in empty holes?
There is nothing listed there for a relay (in red).
This connector is under that fuse block, there is 2 different designs, but it's not under the dash, it's a bulk connector that the check engine light control wire runs through.

I'm not sure, there isn't a picture of it, for a location it just says under the fuse block. But I do see that of the 2 ECM connector locations you unplugged, the with the black plastic with the pins in it is the C1 connector that has the Check engine light control wire in it. If you want to skip that connector under the Fuse block, and check the ECM C1 connector, you will have to back probe the connector with a T pin and check with a multimeter what the circuit does from key Off to key On.
You might have to remove the outside cover of the connector to access the back of the wiring. But to test it, it will have to be plugged in, it's that or unplug the cluster harness plug and check it from there. It looks like it's only 4 screw to pull the cluster out and check it for power across 2 pins of the cluster connector. The 4th diagram is instructions for checking the power feed and ground to the cluster. You can check pin 18 for continuity to ground when you turn the key on for the next test. Pin 18 should be grounded by the ECM when the key is turned on and that's how the check engine light comes on. But I have a feeling it might be a circuit issue going to the ECM or an issue with the cluster itself. Just let me know what you want to test.

If it has that function, yes. Every aftermarket scan tool is different. That pin 68 should go to the ground when the key is turned on, if the bidirectional controls allow you to turn it on and nothing happens, you'll still have to diagnose the circuit. You can also try scanning the Cluster for any codes if you have access to every module now. There should be live data PIDs in each module that you can look at. But there's only really 3 possibilities, the PCM is not grounding pin 68, the cluster LED is burned out, or there's a circuit problem. Since you have other lights on the Cluster it's not an ignition voltage issue. I would suspect a pin fitment problem, which happens a lot when it gets to be summer. If you think about it, your car is sitting in the sun all day, the dash gets super-hot and these issues show up, we see a lot of different things when seasons change.
If you decide to back probe the PCM connector to check the circuit, you just have to be extra careful when doing it. If you're nervous about it, check the connector on the back of the cluster instead. It's only the 4 bolts.

The recommended test for the MIL is at the Cluster connector, which really is not as difficult to pull out as you may think, once the 4 bolts/screws are out, the Cluster will pull right out and you can unplug the connectors at the back, Service info wants a test light in place of the MIL. And then command the MIL on with a scan tool and see if the PCM grounds that wire at the Cluster connector.
The other way, testing at the PCM connector, you will have to back probe the C1 connector on pin 68 while it's plugged in, (this can be the difficult part). There is usually a plastic cover on the PCM connectors that will have to be removed for testing, because you need access to the back of the connector's wiring.

This plastic cover will have to come off of that Black connector, the one unplugged in your picture, Thats how you access the wiring, then you will have the wires exposed (diagram 2). And find pin 68 (6th diagram) is looking at the connector face, notice where pin 68 is, to the bottom of that notch in connector C1. It's a brown wire with a White stripe.

Diagrams 4 and 5 show some back probing pins, they have to be small enough to not damage the connector pin inside, You're just basically sliding the pin into the connector following the wire, not jamming into the wire.

You can pierce the wire instead if you're more comfortable doing that, but you have to seal up the tiny hole afterwards. There is an example of a wire piercing tool in the 7th diagram, it's just making contact with the copper strains inside the insulation.

Either way, with your multimeter hooked to battery negative and the other lead touching the PCM wire, meter on DC volts. Once you turn the key on, the meter should read close to 0 volts, because the PCM is now closing the circuit to Ground (battery negative), And that's how the MIL comes on, the PCM just gives it a ground. The power feed comes from the Cluster ignition feed wire. If you read 12volts the entire time, then either the PCM is not grounding that pin, or there is an open circuit from that PCM pin 68 to the Cluster. Or the PCM has a failed circuit inside and is not able to complete the Ground circuit.

The test at the cluster is easier because you just need to unplug the connector and put a test light or multimeter in series with 2 pins of the Cluster connector and turn the key On. You won't have to be under the hood at all for that test. Diagrams 8 and 9 show how you don't need to really even command the MIL on, it should come on at just the key On, but the scan tool command on test will test the PCM driver for the MIL.

Just don't do any testing that you are not 100% sure on, you don't want to short the PCM out, especially since the vehicle runs, and this is just a light test. I would be more worried about the PCM back probing test than pulling the cluster, I have seen people cross pins while back probing connectors and shorting out the 2 pins together,
You can just leave it the way it is and scan for codes every once and awhile if you're not comfortable on these tests, I'd rather you do that if you're unsure.

You're just putting a test light in between 2 pins of the Cluster for each test. Then turning the key On. Thats all. If the PCM doesn't provide a ground, there's a wiring issue on the control wire or a bad PCM. You can look for videos.

Just don't spread the connector pins on any connectors, using back probing pins on connectors is usually safer so no female pins get spread causing pin fitment issues afterwards.

If you end up having to track this issue back to the PCM, you'll need back probing pins for other connectors as well. I don't think you're going to find the Cluster is missing the ignition feed since everything else works on it. So really the test #2 for pins 5 and 18 is the one that will tell you if the PCM is grounding the MIL bulb.