1987 ford temp automatic - will start when engine is cold, run until it warms up, approximately 5-10 minutes, then just shuts off.

Hmm...........
1985..........
Probably carb? Automatic choke not working? Remove air filter and spray carb with cleaner, Make sure all parts move freely. With engine cold, check automatic choke position. Start and check function as it warms up.

Possibly an idle air motor for your idle problem or a vacuum hose off.

bad bearings on crank or worn oil pump. Pressure will be up when cold, but will drop when engine warms up. Check for knock in lower end. If so, bad crank bearings.

Your vehicle uses the coolant temp sensor and the intake air temp sensor to determine the engines temperature on a cold start. If one of them is dramatically different than the other it will flag a code.

Is the check engine light on?

When was the last time you had a tune-up? if recently what brand of plugs where used?

Have you noticed any hard start issues after a hot soak? A hot soak would be driving the car until fully warmed up, shutting of the engine for approximately 15 to 20 minutes, and then starting the engine again.

I’m happy to assist further over the phone at https://www.6ya.com/expert/jeremy_d728a59f986299fa

Ok, time for car school. The vehicle is operated by the computer, every electrical concept that is. When the engine is cold, it starts fine because the computer has placed it in "cold start mode", putting more fuel into the engine than normally would until Once the engine is warmed up, a temperature sending unit or temp sensor tells the computer to change to normal run mode. IF said temp sensor is faulty and is not telling the computer to make said change, it stays in cold start mode, still dumping excess fuel in the engine that is now not required, cause it's warmed up. Like keeping the choke closed on older vehicles. Now I think you can answer your own question, when this no start action is happening after warm up and it always starts fine when cold, what's the first thing you check ? Right, the unsuspected temp sensor. Change that and no more start problem. Have a wonderful day. Glad I could be of service.

check engine coolant temperature when it gets warmer, or at operating temperature, and verify that the temperature gauge is showing warmed up to operating temperature. If the coolant temperature sensor tells the computer the engine is cold it will over fuel and miss and the engine will die. When cooled down it will run fine until it warms up. Also could be coil if it starts missing after warmed up.

after it has shut down check for gas at the fuel rail with the key on their should be gas squirting out constantly it sounds like the fuel pump is going out

Replace the glowplug relay.Its faulty.The glow plugs stay on until engine warms up.The engine temp switch then activates the relay to switch off.So it could be the temp switch also faulty.

Check to make sure your thermostat works right by monitoring the warming up of your engine. With a completely cold engine, start and run the engine for a few minutes, then put your hand on the top radiator hose to monitor if and when it gets warm as the thermostat opens more and more and note the gauge reading( if it works) the top radiator hose should be unbearably hot within 5-8 minutes. If the therm. opens right it should close right aswell, staying open a little while eventhough the radiator may have cooled may be what you are experiencing causing the rapid cooling and is nothing to worry about. The Hose should not get warm very soon after starting the engine. Regarding you gauge, lets take one step at a time, verify proper op. or the therm.

Check your coolant temp sensor. It is thinking that the engine is cold when it is actually warm or warming up and is probably not giving the correct air to fuel ratio.