Backfiring and skipping

I would also suggest to look at the manifold vacuum. If it fluctuates then you have a burnt valve. So a running compression test would be a good idea to identify the bad cylinder.

Does this have a 4bbl carburetor or is this that tuned port injection (TPI)?

Are the back barrels opening?
Actually the popping out the tailpipe indicates a too rich fuel mixture, so the back barrels would have to be either opening too soon or dumping too much fuel when they do open.
I am not that familiar with the workings of an edelbrock carb to which if they have separate jets fro the front or rear but I was just asking about the fuel system cause it seems to me as though that is where the problem lies simply because of the fact that the problem isn't there at idle or just off idle but rather when at high rpm.
Anyhow on the exhaust valve not seating, I learned a trick years ago to test for an exhaust valve not seating, but I think it can only be done at idle cause it involves stretching a crisp new dollar bill across the tailpipe like you were taping someones mouth shut and hold it tight, and if an exhaust valve is not seating fully every once in awhile the dollar bill would try to suck in.
Another thing on checking for a misfire is to use a temperature gun on each exhaust manifold runner to see which is cooler than the others.

Try resetting the computer, there may be a chance it is picking up the wrong signal from the engine. You can have it done at most auto stores and up until 1985, if I remember you can do it yourself. Disconnect the positive side of the battery, take the wire and ground it to the body or a good ground. Let it sit for a minute. Then reconnect it to the battery. Grounding it out will clear any codes. Notice you will have to drive it about 50 miles before the computer completly resets its self. The 305 5.0 has a computer number 16060631, that is for the carburated engine

Spits back through the carb is exactly the opposite of back firing out the exhaust pipe.
Backfiring out the carb is usually due to too lean of a fuel mixture, or crossed ignition spark plug wires, a blown head gasket, timing chain that jumped, intake valve not seating.
If there is any fireballs coming out the carb air horn when snapping the throttle open and I suggest you remove the air cleaner, then either crossed spark plug wires, an intake valve not seating fully or a blown head gasket. If like a white mist backfiring then timing chain jump. If just popping back out the carb without any mist or fire, then a too lean fuel mixture.

Stand back there by the tailpipe and smell. Does the exhaust burn your eyes? If so the engine is running too lean. Let the exhaust from the tailpipe blow onto your hand, then smell your hand, does it smell like gas/ If so, you are running too rich. If this has 2 tailpipes and a true duel exhaust, meaning no y pipe feel is both exhaust gases are hot, if one is much cooler than the other that is the engine bank with the problem.
Also at idle listen if there is any popping or spitting any fluid out either tailpipe.
Perform a cylinder balance test to find out at least which cylinder or cylinders are misfiring. Then remove the spark plug and smell if the brand new spark plugs already smell like gas meaning it's running too rich or if they are already covered in motor oil meaning you either have a problem with rings or leaking valve seals. To do a cylinder balance test wear some thick gloves and use insulated pliers and remove each spark plug wire one at a time at the distributor cap and put it back on and listen for the engine to misfire, stumble and rpm drop slightly, meaning that cylinder is working.

The ECM is under the dash on the passengers side. There are two screws that hold it up in place.

Does this even have a feedback carb on it?
Have you tried unhooking the ignition timing wire yet, to knock the ecm out of the equation, so the ecm will not be adjusting the timing? If not, try that.
Unless this has just a drop-in hei with no ecm even hooked up to it.
Also, if this intake manifold has an egr on it, make sure the egr isn't leaking or sticking open, at idle reach under the egr and press the pintle open, the engine should stumble or die, if not then the egr is leaking or stuck open.

Then I don't see how the ecm would be affecting it, even if it does have one hooked up.

And yet another possibility is that I am almost sure this originally had an egr but with that edelbrock on there and the fact that you said you didn't think or know that it had an egr might mean that it don't. An egr actually helps cool the combustion chamber cause the exhaust gas is a lot cooler than the combustion chamber and engines that are designed to have them usually run hotter combustion temperatures if the egr's are removed. With that you get the pinging and knocking associated with the excess temperature so people usually set back or retard the ignition timing in order to get rid of the pinging and knocking but then with that done the idle is too low, so they crank up on the bae idle. Now what happens is the carb throttle actually uncovers the main metering section of the carb and the carb begins to drip raw gasoline down the throttle bores and you will get that spitting and sputtering, gas fouled spark plugs, misfiring and popping out the tailpipe. As easy way to tell if the carb is dripping like that is to look down the carb air horn once the engine is warm, choke open and idling and look, with a mirror and flashlight, so it don't backfire right in your face and see if gas is dripping on the throttle plates. If so and if you don't want to purchase an intake manifold with the egr hole, then you could install a curve kit in the distributor, changing the springs and weights and also try to setup the timing just at idle, by either switching the vacuum hose to the distributor from vacuum that isn't present at idle to vacuum that is and probably right from either the intake manifold or I believe on the edelbrock there are your choice of ported and non-ported vacuum fittings right there on the front. Or an adjustable vacuum advance kit. You may have actually more than just 1 problem with this engine so better check out all the possibilities.

Engine performance issues are another symptom of a possible problem with the ECM. If the ECM has any issues, it may throw off the timing and fuel settings of the engine, which can negatively affect performance. A faulty ECM may cause the vehicle to experience a reduction in fuel efficiency, power, acceleration. Back firing through the carburator or exhaust. There is a spark problem, backfiring through the carb, usually mean a bad valve weak intake valve spring, or crossed spark plug wire. Backfiring through the exhaust can mean the same thing except that a exhaust valve may be bad. Take a compression check. Don't rely on doing it once but a couple of times. If there is a 20 pound difference in any one you might have found the problem, but I still think it is a timing issue. There is one subject we had not discussed, check all the vacuum lines. Sometimes they look good but underneath there may be a small crack. Rubber hoses rot over time so check them out.

I'll interject something stupid!

Check Mr. Distributor's mechanical advance weights, maybe the spring that draws them back in broke. Maybe they are stuck?

The Medic