2004 explorer starts then dies after 10 seconds. Won't start for another 10 minutes or so and then starts once and does the same thing

The fuel pump is suspect but I believe there's also a fuel filter inline on the underside of the drivers side chassis. Fuel works off pressure so if the filter is clogged, the pressure remains between the fuel tank and the filter. Once the fuel has siphoned off between the filter and injectors, it dies but because there's positive pressure at the fuel pump, it won't come on until the pressure slowly seeps past the clog after hours. It's worth a look!

I saw what you replaced, didn't see any testing? What testing have you done? Make sure battery has a full charge?
When the problem is ongoing, have a helper crank it, while you check for spark at the plugs. If spark everywhere, use a gage and check proper fuel pressure. Also, check fuel injector pulse. Any applicable trouble codes?
If you have the spider fuel system, that system is noted for problems.

You may be having an intermittent failure with the fuel pressure regulator - checking the fuel pressure when the truck won't start may pinpoint this problem. If the pressure looks OK, this same problem can be caused by the ignition system as well (ignition coil or control module).

The issue with the key is because the car battery ran down. This can be compounded if you jump started it.

try replacing your ignition module the grease that keeps it from overheating has probably dried out

Is it out of gas? Does it have spark? does it have fuel pressure? Is the engine overheated? There are MANY things that can cause an engine to die out. A LOT more information is needed. Nobody can help you with your problem until some proper diagnosis is performed.

Check your fuel injectors

Im assuming you are refering to the anti theft security key relearn process.

1. Perform a security system relearn.
2. Let the vehicle sit for more than 10 minutes with the key off.
3. Attempt to start the engine, leave the key in the run position monitoring the security system indicator.
4. After 10 minutes, the security indicator will change state.
5. Turn the key off and wait 10 seconds.
6. Attempt to start the engine. If the engine stalls, wait another 10 minutes with the key on.
7. After the second 10 minute timer has expired, turn the key off for 10 seconds.
8. Attempt to start the engine. If the engine stalls, wait another 10 minutes with the key on.
9. After the 3rd 10 minute timer has expired, shut the key off for 10 seconds.
10. The security system will relearn on the 4th starting attempt and the engine should start and run.

If that doesnt work, look into this link http://vatspasslockpasskeysecurityhelp.yolasite.com/

These trucks are known for fuelpump problems. the first thing to check is to see if its got fuel pressure, try sparying starting fluid into throtlbody and see if it starts and runs for a few seconds, if it starts then you will know its a fuel problem, it will die out once the starting fluid is burnt up but at least it will tell you its a fuel problem and if it does start i would bet its the pump. good day.

if this a fuel injected car just check to fuel pressure first.