Did new fuel pump & new coil, still having same problem. can not diagnostic trouble codes. When jumping A & B at the DLC, engine light does not blink. ECM? Distributor?

Take a 12volt trouble light put it on the coil neg.terminal does the light blink when running the starter. If it doesn't blink the pickup in the distributor is bad. Read until the end.
I would check some relays I think they are by the firewall. I would also check the grounds for the ignition system. They should be on the left front head and alternator mount.Advanced auto has a module tester and checks them free. With your fingers on the relays you can feel them operate when somebody else is trying to start the car. On a car with points the coil grounds and fires by the points closing. On the newer models the pickup in the distributor opens and closes grounding the coil causing it to fail. Around 1996 a camshaft position sensor opens and closes firing the coil. An analog volt meter will go from zero to one if you check any of the 3 items. These items are a major cause of no start. The crankshaft position sensor monitors the camshaft position sensor Either of the two cause major problems.
This problem is probably a oil pressure sensor relay at the back of the motor if it doesn't close it shuts down the fuel pump relay.

I have a 1995 Caprice and have noticed if you disconnect the Mass Airflow Sensor(a two wire connector in the rubber portion of the air intake) that it will do this. That is, it will start, and die immediately. I wish my problem was this simple.

On that area of chevy I noticed a mechanic used the plastic handle of a screwdriver and tapped the mass airflow sensor. I watched him do this about 8 times and a rebuilt mass air flow sensor cured the problem. If the engined smoothed out when he done this he replaced. I have replaced many modules in the distributor but could feel hoe they acted. Advanced auto will check them free. I had a tester but had to heat some up with a hair dryer as they checked fine cold. Usually with a module you would be running and the motor would just shut off. It would restart after about 1/2hr.
If it doesn't idle correctly spray along the intake and around the base of the throttle body. Pinch off the hose to the brake booster with channel locks.
Fix any vacuum leaks. The carb cleaner will make the engine run smoother.
To check these fuel pressure regulators find a section of rubber hose and pinch it off with vise grips. If the engine runs much better get a rebuild kit for the pressure regulator and throttle body.
Reply if it is an electrical problem and state the codes.
A 1996 starting circuit has a camshaft position sensor to fire the coil a crankshaft position sensor which goes to the ignition control module for timing signal. If unpluged they should read zero to one with a ohm meter spinning the engine spinning. The tach signal goes to the coil.
Check for loose grounds around the head and alternator frame. Left rear of engine block.

A '95 Caprice will have either the 265" or 350" V-8 (L99 or LT1). Neither of those have a cam position sensor, per se. The engine position is encoded by the optical array in the distributor (OptiSpark). 1996 engines have a crank position sensor, but still use the Opti fo encode engine position for the PCM.

I'd throw a voltmeter on the ignition primary and see if the HEI switching module has power in RUN as well as CRANK.

I would avoid using starting fluid to enrich the mixture. Concentrated ether will tend to cause detonation and can cause damage. Further, spraying into the throttle body would require removal of the intake bellows, which would take the MAF out of the intake stream and cause more problems.

I was thinking of the older fuel injection with two injectors on top. Look at your wiring harness around the coil and pickup coil on the distribributor.
Does the wiring look like a pin cushion from somebody checking it the past or the wiring skinned for meters in the past. I have found a few where a part was changed and the wiring was pulled to disconnect it in this case a wire is pulled out of a terminal and just touching. Normally can be found when wiggling the wire but not pulling on it. I have found this on newer GM's than yours. Also a plug could come loose I found one at the neighbors when walking the dog I noticed he had been working for two days on his ford truck
and changed a fuel pump and ignition parts. I thought he had a bad pickup and was going to check his coil neg. to see if it was blinking with a test light and his truck started up and ran good. His was doing the same thing as yours. It happened to be a plug with a broken lug and was loose. He told me he used a plastic strap to hold it in place.