Sounds like it MIGHT be a wiring problem. At least that's where I'd start looking first. Perhaps when your pump "burnt out" it took some wiring with it. Maybe the pump didn't burn out at all but rather a link supplying juice to it did. Sounds like your primary ignition wiring is good to fire it up but when you let the key drop back to position 2 there's no juice there (either to the ignition or to the fuel delivery system) to keep it running. The backfire may be just a build up of unburnt gas from the previous starts and sudden shut-off's.

All just conjecture of course on my [very amateur] part, but I'd start by investigating the fuel pump relay and wiring associated with fuel delivery and ignition, particularly the fuel pump circuit.
Get the old pump if you still have it and connect it to a car battery and see if it actually is bad. If it still works that would indicate the original problem was something else. But even if it doesn't work, and having been replaced, there may be another fault in the fuel delivery or ignition circuit that was part of the original problem. Check the fuses particularly for the fuel pump. Even if it looks good, spin it in place to clean the contacts, then see if it starts. If not, maybe someone else here has other ideas or could help you with checking the pump relay and/or testing the circuit and wiring to the pump.

Of course I could be totally off base, but from your brief description, that's where I'd start investigating.