It is normal for many vehicles for the fuel gauge to stay on when the engine is switched off. This helps a driver to know how much gas they have remaining without turning the car on.
Bad gauge Bad float in tank
The fuel gauge sending unit in the fuel tank is probably the problem.
Did you check the fuel pressure? You may be running out of fuel. There is enough pressure in the fuel line when you have the key on, and the pump is pumping but as you use up the fuel in the line the fuel pump can't keep up and you lose pressure and the fuel shuts off. Get a fuel pressure gauge and hook it inline on the fuel rail and see what pressure you have with just the key on, and then start and run the engine and watch the gauge. If the gauge drops, your fuel pump is likely on it's way out.
This information can be found in the owner's manual
If engine is running then the answer is yes. And pressure will stay in the fuel line for some time after you shut the engine.
It's likely the float sensor in the gas tank is malfunctioning. It might be part of the fuel pump assembly.
The most likely reason is the sender wire (that runs from the sender unit in the fuel tank) is earthing some where along its route.
It takes 3 things for an engine to run. Fuel/Air/Spark. Which one of these are you missing? You say it runs but won't stay started. Do you mean it starts, runs, and then stalls. If so, I would suspect a fuel delivery problem. Clogged fuel filter, or defective fuel pump or fuel pump relay would be suspect.
Connect a fuel pressure gauge. There should be a test port somewhere near the injectors. Watch the fuel pressure. If it drops just as the engine dies, you probably have a problem with the circuit that keeps the fuel pump running. When you are starting the engine the fuel pump turns on for a few seconds. After the engine is running the fuel pump is allowed to stay on. It depends on how the manufacturer implemented the circuit, it could be a relay, fuse, loose connection or who knows what. Have you fitted the new filter the correct way The direction of flow should be marked on a new filter.
Yes, that's how fuel is delivered to the engine.
If it drops to zero it is the stepper motor in the cluster. If it drops down several degrees then it is the thermostat and will eventually make the check engine light stay on. you will experience poor fuel economy due to the computer loading the engine up with fuel thinking you are in warm-up mode. We have lots of info at our enthusiasts forum at www.trailvoy.com