Starts fine one time and then refuses to start.

I would suggest you start by checking the fuel pressure at the fuel rail. You should have around 55lbs. but need at least 45lbs. for the engine to operate.

adjust the gap to the minimum will solve the problem.

If the sensor is in the dist, the dist shaft could be worn and slowly damaging the new sensor.
Could also be a new problem.

If it has a manual tranny it sounds like a worn out clutch and bearing problem

Next time it does this check the fuel pressure.

I would say yes, if it wouldn't let the car run, then yes the wire may be frayed now, repair the wires first and eliminate that problem . then have the codes read,

If you had a bunch of cracks in your distributor cap it may have caused the pick up coil under the distributor cap to go bad and cause this problem.

check ur timing -
MARK with a scratch tool or use a yellow majiuc marker to see where the timing is now then-

loosen the distributor bolts - if you have a distributor on that model and - turn the distributor one way - then the other and see if it improves. You can slightly tighten the bolts or carry a wrench with you and drive the car and see if a new position improves the running of the car.
Also - change the plugs if you haven't done so - check them to see how they look anyway.
You may have a flooding condition on your hands as well but that is a whole new thing - so stick with timing and plugs -

hope this helps
Gary

There is a condition called thermal breakdown in ignition parts( ie: spark control modules). Pretty common really in older stuff. Why does it happen only after shutdown? because of a condition called heat soak. underhood temps rise higher than ever right after shutdown. Your ignition module is mounted to the side of the distributor. If practical when the problem happens pour cold water on the module give it one minute and see if it starts(wont damage it) if it starts right up you proved it bad. Hope I helped Thanks

Main Fuel injector relay is probably the culprit. Buy a second hand one for about 50 bucks and replace it yourself within 30 minutes.