Pontiac Repair: 2000 Grand AM Engine Stalls, climbing a hill, electronic operation

Question
Harold,
I drive a 2000 Pontiac Grand Am GT with approximately 95,000 miles on it.  This past week while on a long trip, the engine started shutting down while driving.  No instrument lights would come on warning me, and all fluid levels are fine.  This would happen time after time without warning.  Each time I would pull over, let it sit for a 10-15 minutes and then it would start up and run fine again.  It appeared to do it more often under the following conditions: 1) climbing a hill  2)when the transmission changed to lower gears on a grade climb  3)when the gas tank was closer to being empty than full.  I don't know if it's electrical or mechanical.  I'm not sure if I'm having problems with a fuel pump or what.  Whatever it is, after sitting for a few minutes, it will start up and run for a while.  One time I might get 85 miles, the next time maybe only 20.  Is there a thermal couple or resister on the fuel pump that can heat up too much and shut everything down when it overheats, and then be okay again after cooling off? Would lack of fuel shut the entire engine down without warning when I'm traveling 70 miles per hour?  Is there a link between the transmission changing to lower gears and the electronic operation of the engine?  If the fuel has been stopped, what warning indications should I get to know I've run out of fuel?  When the thing shuts down, it literally just shuts down. Everything goes off?
Fuses somewhere maybe?  How would a person go about eliminating things to narrow it down without giving someone a blank check to just rack up costs trying to figure it out for me?  I appreciate your time and your response.  Thank you.  

Answer
There are alot of things that can cause the car to stall, and since you have to let it sit for a little while before it restarts, the most common things are a fuel pump, ignition module, and PCM(computer). All 3 can be affected by heat and by a trans downshift and heavy loads like climbing hills. All three situations require more spark, more fuel, and high speed computer adjustments. There isn't any kind of thermo-limiting switch on any of the parts.
This should be an easy one to diagnose. All you need to do is wait for it to shut down, and then check for spark, fuel injector pulse, and fuel pressure. One or more of these three things will be missing. After you find out which thing or things are not there, let me know and I will walk you through a more specific diagnosis. Or you could take it to the dealership to have it diagnosed.