I did get the coil and plug wires changed. It seemed to make it easier to start, but the hesitation is still there.Did the Plug Wires fix the backfire and hard start issues?
Did you do a compression test? Make sure all of the plugs are out when you do. That way if you have a head gasket failure between 2 cylinders it will be easier to detect.
Common Things to check...
Fuel Pump. ( This is the sneaky one that gets past a lot of mechanics. )
Temp Sensor for ECU.
Computer control systems rarely cause a backfire with the exception of The Temp Sensor for the ECU. Or the ECU Being Bad. Rough Idle and backfire are usually cause by Valve train issues (Worn Valves, Sticking Valves, Worn valve Springs, Worn Timing Chain), Head gasket blown between 2 cylinders, Or incorrect fuel air mixture.
I have seen 500K miles on a couple of stock daily drivers but they were not a Mustang. One was an Acura and the other was a Chevy Pickup.
Original Clutch with that many miles is impressive. Sounds like you take really good care of it.
Lower end is probably getting a little sloppy with that many miles. But if it was to the point of being an issue you would know it. It would be making a lot of noise when it was cold and kinda clear up once it warmed up. Or you would be knocking and ticking a lot.
I have not checked the compression yet. I will either rent or buy a compression kit. with the number of cars I have, I will probably need it often.
I will replace the throttle position sensor, and maybe the ACT. The TPS is not original, but the ACT is.
On cold starts, and when I change oil, I don't hear much from the lower end, so I think it is OK for now.