No ignition spark when hot

So many codes/so many faults all at once makes it seem like some disaster has befallen your car.

I suggest you erase all the codes and see how many come back after the engine has completed a complete warmup/cool down cycle.
Often fewer codes will return and it is only those that do return you need to check out.

So many faults suggest maybe the ECU is telling lies and could be faulty but replacing it is a last resort so the other stuff needs checking thoroughly and systematically first. Use a good scanner to examine live data and verify every diagnosis before spending any money.

Sounds like it might be losing spark. You could test for that. If that's what it is, could be the crank position sensor. Or some other part of the ignition system that shorts out when it gets hot. Ignition control modules do that, so do crankshaft sensors when they fail. Check for spark soon as it dies or stalls. If it was getting both spark and fuel, it should start right back up, immediately. If you have to wait for something to cool down first before working again, could be spark related, but also fuel pump motors can act like that, so take nothing for granted. Check for spark and fuel-you will find the problem.

A couple of things maybe: the ignition control module, or the crank position sensor. I f you have a distributor, the ICM would be inside the dist. If a coil pack, the ICM is probably underneath the coils on the coil pack.
Sometimes, an ignition coil too will get hot and short out. That's 3 things you could check. Have you tried to check for spark when it won't start? If no spark, any of the 3 things noted. If you do have spark, see if the fuel pump is quitting and putting out no pressure.

This sounds familiar, did I answer this last month? It sounds like an electrical connection that gets hot in 5 minutes, opens due to the heat and takes an hour to cool off before the connection will return. It probably continues to "cycle" like this forever untill the bad connection burns bad enough that any amount of cooling will not return the connection. This could be happening at a connection, or a relay. I bought a laser thermometer recently, Home Depot for about $15.00 I recommend you borrow or buy this tool to find the hot spot. or follow wiring from ignition switch to coil through ECU or Ignition computer to find a potential hot spot by feel. Infared goggles may help find the hot spot. Good luck, this is a difficult puzzel and will take some persistance to find the problem.

How old is this vehicle and make and model as well please

Could be a problem in the ignition circuit. Check or test coil and ignition module-one of them may be shutting down or shorting out when they get hot.
Verify by checking for spark at a plug wire immediately after it shuts off: if no spark, it is an ignition problem. Besides the coil and ignition module, other possibilities are the crank position sensor, cam position sensor, ecu, the ASD relay (automatic shuit down relay-it's in the power distribution box under hood). If that relay fails, all power is cut to coil and fuel injectors, and maybe the fuel pump also.
Good luck.

Often when this happens something is failing electronically and can be hard to locate which part. If you know all of the parts that are in the sytems and there location(like the ignition amplifier, the camshaft position, ecm, etc it will be easier to locate the failing part. Often the ignition amplifier, camshaft position sensor and not so often ecm will cause no start when hot. There are many things that can do it other than those few named. When the vehicle will not start it may be often missing spark because of the ignition amplifier. Say it fails and you find out that it has no spark once it is hot. Find that component that you believe or think is suspect and spray it with a can or "Kool mist" or "component cooler" which will cool the electronics down. If you have the correct failing component the vehicle will restart once cooled and at that point you simply replace that component and its done. The spray coolant may come in different names but it all does the same...cooling components or circuits. Radio shacks cooling spray is called "component cooler". When you are spraying the electronics to cool it down, spray it for as long as it takes to cool it and that depends on where it is located; normally just a few second burst of spray will cool most components. This cooling will only help if you have located that failing component. If the vehicle will set for a while and then restart when it's cooled down but will not restart when you use the cooling spray on it, you more than likely just have the wrong component. If you know where the suspected components are, you can most likely find them.

I would check spark, specifically thinking the ignition module is going bad.

Check the prrimary and secondary resistance on your coil. Also check the resistance of the plug wires and plug gap.

it could be your coil. if it is shorting out internally