Eccessive exhaust temp.

Hi.it can be a lot if things.coolant temp sensor wil be the first thing to check then check for vacuum leaks at intake manifold

probably find that the ECM has gone into limp mode to protect the engine
run the fault codes and fix any fault then reset the ECM

Raw unburned fuel is entering the exhaust system. Is the vehicle running rough/misfiring? IS the check engine light on/flashing? If so, have the codes scanned to isolate the problem cylinder causing this issue

run the fault codes as misfire can be from injectors , low fuel pressure and ignition problems
all temp gauges will show a rise on stopping an engine as there is still residual heat in the block and heads with no water circulation and cooling
the fire from the exhaust indicates an exhaust valve is not closing properly so a compression test would be in order to check for valve problems or hydraulic follower problems

you probably have your finger on the problem. have oil pressure test .when engine hot minimum 12psi.and change the thermostat.Don't cost much

yes i agree w/ the other suggestion try higher octane fuel

Check the Fuel Filter and pump. It sounds like you have an over heat problem in the fuel pump unit or air in the fuel line that is causing vapor lock. This can happen when you let the gas level get below 1/4 tank depending on the temperatures in your area.

Sounds like you might have a plugged or bad catalytic converter. Make sure the spark plug wires are in the right place.

If it is carb version check if auto choke is getting power. If fuel injection replace temp sensor (common fault).