What are the possible problems if my 1999 Ford Taurus' engine begins to overheat after running after five minutes?

Go to an Autozone, Oreily, or Advance auto parts store. They will scan you car for free and can reveal the Code and possible fix.

Possibly the air sensors went out of balance when there was no fresh air. Sounds like sensors read the engine as "Lean" due to lack of Oxygen and set engine to a rich setting.

Do you know for a fact there is no air in the cooling system and it is full of liquid coolant ? Are the fans coming on with the ac and when the engine reaches 220 degrees ?
Is it possible the water pump is failing internally ?

P1131 Lack of Upstream Heated Oxygen Sensor Switch Sensor Indicates Lean...

P1132 Lack of Upstream Heated Oxygen Sensor Switch Sensor Indicates Rich...

The blinking engine light indicates your problems are severe and could cause more damage to the engine if not corrected. You will need to have a diagnostic scan run to retrieve the trouble codes that are causing the light to come on & flash. From those codes, repairs needed can be determined.
I hope this helps.

Making sure the fluid level is up, and there is no leaks in the cooling system, then you must have a circulation problem, Check the thermostat and water pump.

blown head gasket would be my first thought but i would check the water pump, radiator, hoses and thermostat before removing the head

Check the fuse for the fan and you could test to see if you have a voltage running to the fan when the car starts to overheat. If you have no power to the fan then trace the wore and see if you cant find the problem. If you have power then the fan motor is most likely bad.

You will overheat the engine if losing coolant.Firstly replace the cap with a new one,Is the engine temperature normal when you stop ? If its over temperature it will still blow out coolant. Check the radiator for blockage of the tubes & ensure the fan belts and water pump are in good order. Worst case is a blown head gasket can test for combustion gases in the coolant

Sounds like a stuck thermostat no heat water not circulating to heater core engine overheats stuck thermostat won't let water circulate through engine