From the OPs description the engine rotates freely from the starter but with all spark plugs in and connected he gets a pushback and a pop out of the carb. the pushback seems more like a cylinder firing way too early and the pop out the carb is another clue that the spark is happening when an intake valve is open.
I wish the OP would answer some of the questions I have asked. The answers help guide to a solution.
- When cranking from the relay is the key in the run position? (car will only start if the key is in the run position when cranking from the starter relay, and it might not start if there is too low a voltage to the coil due to no bypass on the IGN2 cir)
- Was the key in the off position when it cranks freely from the starter relay with a plug out?
- How do you know that the distributer is timed properly?
- Was the dist pulled to install the protronix?
- Is the firing order rotation direction correct? BB rotates CCW, SB rotates CW
- is there 12V (nominal) on the ignition side of the Ballast with the key in the run position
- Is there less than 12V on the coil side of the ballast with the key in the run position (6-10V)
- is there cranking battery voltage to the coil side of the ballast while cranking (cranking battery voltage will be less than battery voltage because of the load of the starter, might be as low as 10V)
- did you follow the Protronix wiring diagrams and instructions
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