07 F150 stalls out, won't restart for 5 minutes

I would check the fuel pressure fords have a test port on the fuel rail but you will need an adapter for fords because they use a specific fitting just for fords. You can find the specs online. If you can get a fuel gauge long enough would put it under the windshield wiper and drive it until it does and see if you are losing fuel pressure. My guess is that the fuel pump is bad and it heats up and shuts down and once it cools down it starts working again. You should never run less than an 1/8 to 1/4 tank of gas because the fuel in tank is what cools the pump. And the pump shouldn't keep running so you might have a stuck relay or or a wiring problem. The pump should come on for 2 secs when you turn on the key to prime the fuel system and after that it uses oil pressure to turn on the pump and that's so if you lose oil pressure it will turn off the engine so it doesn't damage it. And if you change the pump always change the fuel filter I would do the fuel pressure test first and then take off the fuel filter and blow through it and make sure you are doing it in the direction of flow and you should be able to tell if it's plugged and make sure when you put the new filter on it's in the correct position for flow it will say on the fuel filter which way you point it. That's what I would do as a mechanic.