Engine runs rough and backfires

What firing order are you using?
Here is the firing order 1-3-7-2-6-5-4-8, counterclockwise and not the cylinders on the passenger side from front to back are 1, 2, 3, 4 and drivers side 5, 6, 7, 8.

I read what you stated, I was just trying to make sure you had the correct firing order, rotation and cylinder identification, so can you confirm the firing order, rotation and cylinder identification?
With the backfiring out the intake and exhaust it sounds like you have a crossfire.
Either than or you have both a too lean fuel mixture, choke open too far and a too rich fuel mixture, dripping carburetor down the air horn on the inside.
That could happen if there is too much fuel pressure forcing the float needle off the seat, so how much fuel pressure do you have and have you checked for a vacuum leak.
Does the engine at least stop backfiring back out the carburetor once the engine is warm and choke wide open?

What are you asking, "Would that be carburetor or vacuum?"
Yes, the compression test should have confirmed if a valve was not seating.
Remove the air cleaner and filter, first of all take a shop rag and while the engine is warm and choke wide open, take the rag and cover the carburetor air horn and shut off most of the air going in, the engine should run slower, maybe stumble or even stall. If it actually runs better like that, that proves there is a vacuum leak. Then if the engine does barely run, try shutting off all the air going into the carburetor. The engine should shut off. If the engine still runs with all the air cutoff, then it must be getting air from somewhere else, such as a vacuum leak.
Now, with the engine idling warm, choke wide open, use a mirror and flashlight and look right down the carburetor air horn and look to see if you can see any gas dripping onto the throttle plates.
You never mentioned what your fuel pressure was and also you never confirmed your firing order, rotation and cylinder identification.

You could also do a cylinder balance test and find out which cylinders are misfiring, by using a think pair of gloves and insulated pliers remove each spark plug wire at the cap and put it back on and listen if while pulling each one you get a drop in rpm about the same for each. Let me know which are misfiring.

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