Is the fuel pump about to fail?

I have a 04 Ford Mustang (automatic). A year ago my car started to lose performance. I have it replaced the throttle sensor, the spark plugs, fuel filter, oil change, thermostat, air filter, but nothing has corrected the problem. Now, my car makes hard gear shifts. it pulls back and forth when it finally go into the next. When entering highways it takes time to get up to speed. No check engine light has come on yet. Now my car makes a light rough idle(5 sec) when I turned the engine on. When I fill the tank, it keeps shutting down the pump even when it's not full. I also noticed that when I turn the key on, the fuel gauge stays on the empty position for 2 or 3 sec and then moves. I havey reasons to believe that the measurements are not accurate at sometimes (ex; it measures 1/4 now, then in 5 min it's 1/2).

What is the problem?