If the theft light is flashing that is why your truck is not running, it thinks that it is getting stolen.
The problem is the system cannot be bypassed or disabled. It is built into too many systems and there is no way to remove it from all the computers. This is expecially true as all the Explorers have the system (as do most Fords) and there is no parts to change them to that don't have it.
Did the vehicle run with the one key at one time?
If the theft light is flashing continuesly, then the vehicle is disabled.
If the theft light goes out when you are cranking over the engine, then yes the theft system is NOT the problem. On your vehicle, I would be looking at the fuel pressure. And yes I know that you have replaced the pump, but that is where I would start. I have put in more then one bad pump in my life time.
Also check the Interia switch, and make sure that it is not tripped or unplugged. It is located on the fire wall, dead center, in the passenger side foot well at the top of the carpet. I have seen them get unplugged by the passangers feet.
Your fuel pump is shutting off. You need to locate the fuel pump relay and using a jumper wire, jump to two large pins. The relay is a small one on this truck and on the bottom there are either two or three small pins and two large pins, you will want to take a jumper wire and go from one parge pin to the other. You are basically doing the job of the relay. Then retry the vehicle.
Here is a picture of the fuse box:
The check valve in the fuel pump is bad. The problem is the only way that you can replace the check valve is with the fuel pump. The check valve is designed to retain pressure inside the line so that the vehicle will fire up quickly (on the first key cycle).
However it looks like you also have a bad fuel pump relay. With you jumping the relay and the pump turning on that tells me that the pump is working but that the contacts inside the relay are carboned up. This normally happens when the fuel pump starts to draw more current then the relay was designed to handle. That is also a sign that the fuel pump is bad as well (excessive current draw)