[ImperialCrown]

Is the purge solenoid fault code still active?
You don't want an overly rich mixture on a warm engine. A cold engine can handle a rich mixture better than a warm one. Try pulling and plugging the purge hose for a temporary 'warm drive' test.

 

[ImperialCrown]

What was the actual fault code? It sounds more fuel/emission system related than secondary ignition related. It's easier to fire a warm spark plug than a cold spark plug.
After a few moments of running the engine will enter closed loop (mixture control using the O2 sensor). Too rich or too lean mixtures can cause stalling. Are the plug tips white or black?
Fuel injectors themselves were known to leak into the venturi on these making an overly rich mixture. Have there ever been any O2 sensor fault codes?

 

[ImperialCrown]

Would 35 be the radiator fan circuit? Does the fan come on?

 

[ImperialCrown]

I wouldn't have the radiator fan on all the time. That is not a good practice.
If you had the evap purge hose off and plugged and it still stalled, then I don't think that the purge solenoid is causing the stalling.

 

[ImperialCrown]

The 2-wire coolant sensor near the upper radiator hose connector has to be disconnected in order to accurately set ign timing.
It will turn on the 'ck eng' or 'power loss' light, but you can erase the fault code after you are done.
Edit: I was under the impression that it was stalling at idle. If it is hesitating or sagging under acceleration, does the engine actually shut off and you have to restart it?
Set the timing first.
This could also be a plugged cat. The air pipe that broke off supplies oxygen to help the cat process the HC and CO. Without the fresh air source, the cat can get overworked, meltdown and plug up.

 

[ImperialCrown]

This can't be done by ear. You need to buy or borrow or pay someone to set timing.

 

[ImperialCrown]

The fuel pressure regulator at the injector should be able to keep pressure around 14.5 as stated. A 'real-time' fuel pressure test would be good if any fuel pressure doubts remain. The pump was checked at the dealer and they said that it was OK, but maybe the problem wasn't present at the time.
Cold engine acceleration would use more fuel than warm engine acceleration. Backfires through the throttle body (pop-back) can occur with lean mixtures and to a lesser extent, with overly advanced timing (it would ping badly and maybe struggle to crank over when hot). Backfire through the exhaust can occur with rich mixtures (i.e.-raw fuel in the cat) or severely retarded timing.
Avoid Bosch O2 sensors and spark plugs if you've used them. Always go with OEM.

 

[ImperialCrown]

As white plastic ages inside a fuel tank, it will turn orange. Can you find where the piece connected to on the sender-side? Did it join to the smaller rubber hose on the fuel pump?
Depending upon the rust and fragility of the fuel tank, can the dent be popped out ? If the seams are already weak, this may start a leak.
This tank may have been shared with other K and E-bodied vehicles. so a good, used and complete one may still be around.

 

[ImperialCrown]

Is the fuel pump unit and the sending unit two separate units in two separate holes in the back of the fuel tank? Or are they combined in one unit in the tank?
You may want to measure fuel pressure when the car is acting up. If the pressure drops low when the car sags and sputters, you may need a fuel pump. Was the inside of the tank clean? Diagnose first.