Engine is spitting and sputtering?

No testing the resistance of the coil primary internal circuit is not good enough.
If you had a backfire into the intake manifold, and the spark plugs came out dry, then there probably isnt any fuel getting into the cylinders, the starter fluid wasnt enough to even start the engine or you didnt spray enough in, either way, a backfire into the intake is most commonly a lean mixture. Meaning not enough fuel, or any fuel at all. The MAP sensors signal is going to be near or close to 4.5v-5v without the engine running, if it was running and there was actually vacuum in the intake manifold the MAP would be reading 1-2volts, at higher voltage its just reading atmospheric pressure. Meaning no vacuum,
It sounds like you have spark if you had a backfire, are you sure you checked for an injector pulse correctly?

Heres the testing of the cam and crankshaft position sensors using a test light and the ignition coil power feed, if you dont want to check for spark, theyre having you check for power remaining on the coil power wire while cranking, if theres no crank or cam signal the power will shut off from the coil, because the PCM will power down the ASD relay if it doesnt see both cam and crank signals, or just watch the video below and check for spark,

Watch this video, go right to 9min 35seconds in the video, he does a spark check using a test light to check that the coil has good output.
https://www.youtube.com/watch?v=yw48gKmsy-0

https://www.2carpros.com/articles/engine-backfires-while-running

So, since your test results are kind of reverse of what they are stating here, they are saying you will only get the one second power on for the ASD circuit if there are no cam/crank signals. And then it should only be a momentary flash. Then the PCM de-energizes the ASD circuit due to no cam/crank signals,
If we were to follow the flow chart here, with the cam sensor unplugged and not flashing of the ASD power feed, their diagnosis is a bad cam or crank position sensor. Or the signal for those sensors is not making it all the way to the PCM, at this point, you can try replacing those sensors.

But you really should check for spark here, you're not going to hurt the PCM, we check spark all the time, whether you use a spark check tool or a test light it doesn't matter, we need to know if the coil is outputting any spark.
I've added the spark check from service info as well, since the spark plugs are coming out wet with fuel, it sounds like the injectors are firing, check the oil too, see if it smells like there's gas it in, it's probably completely contaminated with fuel by now.

You can see in the video I posted of him checking for spark with a test light, he is a teacher at a college, teaching automotive diagnostic testing and one of the best. He wouldn't be checking spark like that if it wasn't safe for the PCM. I do the same testing as well. Since the crank and cam shaft position sensor signals are only a 5-volt output, you won't be able to see it with a test light. You need an oscilloscope to see those signals.

https://www.2carpros.com/articles/how-to-check-for-ignition-spark

Did you have spark on the front 3 spark plug wires and did it jump at least 1/4in?

When you went to do the compression testing, what did the front spark plugs look like, were they dry or wet with fuel, or black and carboned up? And be specific please. Did you check for bad gas in the tank as well? You've mentioned that once, but dont tell me if you checked it. We have been doing this for a month , it should be clear what the issue is by now.

You have spark, you said you had an injector pulse, your fuel pressure is low (should be at least 48psi), you mentioned it might have bad gas, you have plenty of compression, so either something was missed or the exhaust is restricted.

If you have spark then I think the crank sensor is fine,
And if you are absolutely sure you did the injector pulse test correctly and the injectors are pulsing on the Ground side of the 2 wire connector, then we are down to only a few possibilities. But you have to be positive there is an injector pulse.

Im not sure how much starting fluid you used, but I would try that one more time, have someone else crank the engine while you spray either starting fluid or even brake clean works better into an open port on the intake manifold.
But you may not have used enough since this engine has been cranked over so many times, the cylinders are completely washed out by now if theres any fuel spraying from the injectors.

you need to be sure of the results of your other tests so far, and I am not.

That being said, it leaves either weak fuel pressure, where the spark plugs would come out wet and carboned up due to lack of combustion, or possibly a restricted exhaust. Meaning the Catalytic Converter has failed.

It looks like this vehicle only has one Oxygen sensor in front of the Catalytic Converter, on the back of the engine where the two exhaust manifolds meet and there is the one exhaust pipe going to the Converter.

To check for a Failed Cat, you would need either unbolt the front of the Cat to allow exhaust to escape or to remove the Oxygen sensor and crank the engine, with the sensor removed it also gives the exhaust a place to escape and the engine should try to start.
With a restricted exhaust, the problem of starting just gets worse over time until you are left with a no start condition all together.
But since you said the engine is almost trying to start, the converter may not be fully restricted. They fail due to misfiring cylinders, rich mixtures, oil burning, coolant burning etc.
With the information I have, thats where Im at, and I cant be there to see any of the test results, I am assuming the things tested so far were ok, and assuming is not good.

Do me a favor and watch this video, at least the first 5 min, it shows how much starting fluid it took to get this engine to start for a minute or so.
https://www.youtube.com/watch?v=2n1kQBSHb7A

https://www.2carpros.com/articles/car-not-running-advanced

https://www.2carpros.com/articles/bad-catalytic-converter-symptoms

https://www.2carpros.com/articles/how-to-test-a-catalytic-converter

If you need an example of checking back pressure for a restricted exhaust here is a video showing how to do it, I use my vacuum gauge which also goes up to 10psi of positive pressure, and when checking for back pressure at the Oxygen sensor port, there should be no more than 1-2psi at max, even 2psi is too high.
In the diagram it shows where the front Oxygen sensor is located on the back of the engine. Hopefully it comes out easy, but dont force it or you'll stripe the threads out.

You could even check back pressure at the EGR tube coming off the exhaust pipe if its before the Cat, that might be easier, just do not leave the gauge on there long, I have seen people melt the hose and gauge from leaving it on the exhaust too long, but this is a no start condition so you should be ok. I marked the EGR tube in red on the 2nd diagram, if you cant remove the Oxygen sensor, or unbolt the exhaust at the manifold here, the EGR tube would be much easier to get to.

This car still starts but has a bad Cat.
https://www.youtube.com/watch?v=IPEq53z2iE4

You also mentioned the MAP sensor was not properly installed at one point, I assume you got that put in correctly, I would be curious to see what the MAP sensor signal does during cranking. The engine should produce some vacuum during cranking so the MAP sensor voltage signal should drop somewhat during cranking, Lets check that as well since we are checking everything else.
The Signal wire is Green/Red pin 2(middle pin), the 5volt feed is Purple/White Pin1,
and sensor Ground is Black/Blue pin 3.
Signal should be near 4.5volts key On, and drop a little while cranking.

This is the MAP sensor flow chart if you have good spark, good fuel pressure, good compression and timing, and a 5volt feed to the MAP sensor, they have just replace it.

Which Im not against at this point, starting to change some parts is not the correct way, but we have been at this too long,
You can do this before or after checking the Catalytic Converter.

Okay, the reason you didn't see any power to the coil with the cam sensor unplugged is because the PCM will de-energize the ASD relay if it does not see a cam signal directly after the crank signal, so that means that those signals are okay, and all the more so since you have replaced the sensors already. Having to pump the gas pedal to keep it running or try to run, it sounds like you are running out of fuel.

And I understand this diag has been a huge pain, with no codes setting, its all the more difficult. I don't think you have a timing issue here, with the compression readings you have, this is not an overhead cam engine, like many modern vehicles are, this is a one cam engine with pushrods. Newer vehicles have dual over head cams that cause all kinds of problems with valve timing.

The timing chain could be stretched, but to check it you need to pull everything apart and get all the way down to the timing chain itself. You would have to pull the timing cover; this is what they want you to do to measure the timing chain stretch (diagram 1 below). You might as well just replace the engine if you're going to go this far, because the engine is old and has high mileage.

If you want to keep going and check this pump, you can block the return line going back to the fuel tank and dead head the pump, meaning it will put out its max pressure, you will only do this long enough to get a reading, but this pump has a stall out pressure of 135psi,
Without enough fuel pressure the injectors will have a difficult time firing as well, they need fuel pressure behind them to over come the pintle inside the injector.

To add to it, if the gas is old or bad, you see where I'm going with this, a vehicle this old can have multiple problems by now and I'm sure it does. You can try a fuel pressure reading right at the fuel filter so you're coming right off the pumps supply line.

Okay, looking back at your first posts, you mentioned it had been acting up in the past, starting then stalling, then run smooth for a bit but stall out again, are you sure you don't have a tank of bad gas? Did you happen to get gas at a place where you usually don't get it? If there is water in the gas from bad tanks at the gas station, which I have seen plenty, it will send you on a wild chase like this, Bad gas can cause all kinds of weird issues and we end up down a rabbit hole as we are now.

Pump out a couple gallons of gas into a clean bucket, and let it sit for an hour or so and see if you end up with yellow gunk at the bottom of the bucket, It probably won't smell like gas anymore either. With a metal gas tank, I would also question what is inside the bottom of that tank by now.

I helped a friend with this same issue, we chased our tails on that one for a week, it burned out 3 ignition coils, it burned out the PCM, all because he had an entire tank of bad gas from a station he had never been to before. The misfiring cylinders overheated the coils, which in turn, burned out the PCM. Once he pumped out a couple gallons of gas, it was filled with yellow gunk from the water and contamination in the gas tank. All that crap got stuck in the injectors, in the fuel rail, we tested everything trying to figure out what was going on, then he remembered it all started after filling the tank one day. He had to drain the entire tank because it was settled to the bottom, gas and water will separate after a little while in a bucket. The heavier contaminations will also settle to the bottom.
A hard lesson to learn.

Looking through all the technical service bulletins, there is an old one for a revised fuel pump along with fuel tank straps. Looks like a high pitch noise can be heard from the pump in some cases.

Do a couple other checks with your multimeter set on DC volts, with the key On, check the voltage level from Battery Negative to the engine block, and then Battery Negative to the Body/Frame, it should read close to zero volts, this test really should be done running, but with the key On, if there's excessive voltage drop to the battery you should see something above 1 volt.
I see the PCM is right next to the Battery here. If you want to check the PCM connector for any corrosion or pin issues, here's the pinout for that if necessary. Just things to check.

If the spark is jumping 1/4in that should be fine, are you checking the MAP sensor signal wire, not the 5volt feed. You should see a lower number than 5 volts on the signal wire, even at key On, it should be lower than 5volts, 4.5v range. And the fuel pressure is an issue, if there are sections of the windings on the fuel pump that are bad, then the pump is not putting out what it should be, or you have a wiring issue to it. It should go right to at least 48 PSI with the 2 second key on prime, not 20 PSI.

The MAP signal wire is the center wire on the connector, green/red wire. Pin 1 should be the 5volt feed, Violet/White (purple/white). The last wire should be the sensor ground, black/blue wire. If the signal wire is not changing at all when cranking, that's also an issue.
That means there is no pressure changing in the intake manifold during cranking, you should be able to put a vacuum gauge on the intake and see a change during cranking, but a clogged cat will affect that reading too. If nothing is going out the exhaust no air can come into the engine. So, you either have multiple issues going on or a bad wiring problem. You should see around 5in Hg of vacuum during cranking with a vacuum gauge on the intake.

No, the pump has a number of windings, usually 8 or 10 segments, for a brush type motor, and when the windings start to short out, the brushes burn out, or there is a wiring issue to the pump where its and there is low current flow the pump output will be affected. This is why we measure a fuel pumps RPM and current flow using a scope these days to make sure it's the pump that's failed, but they all fail eventually, and with all the key on key off testing, it's probably just worn out, if it's never been changed and its 30 years old, I would be surprised if that is not already the 2nd pump in that vehicle by now. You should have 48psi at key On, that's the prime cycle.

You're seeing multiple faults though, and the more the engine is cranked over, if the fuel injectors are actually spraying anything, the cylinders are getting flooded and washed out. Which overall lowers compression, which is why they are a little low. That's why I keep asking the condition of the plugs each time you mention them being out and if they are wet with fuel. You may want to have a mobile tech come out and at least diagnose the no start issue, and then you can do the repairs. It's difficult to keep track of the state of each system, with varying fuel pressures, possible faulty MAP readings, you can unplug the MAP sensor and see if it sets a code for that circuit.
Technically it should, but during a cold start the PCM is mostly looking at coolant temperature for the main input until the fuel system goes into a closed loop operation. During Open Loop and engine warm up the PCM is ignoring the oxygen sensor as an input, until the vehicle gets up to the correct operating temperature. But if a shorted map sensor is affecting the 5-volt reference, which is in turn effecting another sensor such as the TPS, then the injector pulse is going to be affected, another reason for asking about the plugs being wet. But with a lack of fuel pressure the injectors will have trouble opening as well, and it takes a lot more fuel to get an engine running than it does once its finally running. A cold engine takes much more fuel. There's no way it will start on 20 PSI, there's something going on there. But to get to the pump you need to drop the tank on these which makes diagnosing the circuit more difficult if it's a power or ground issue at the pump itself.
It might only take a trained tech an hour or so to run through his diag process to tell you what's wrong, instead of struggling along for months with this, plus saving yourself all this headache. It takes a long time to learn a process that works and look at the whole picture if that makes sense.