The pump on the frame is the high pressure pump, 40psi.
The pump in the tank was from the old carb days, 7psi.
The old stories about not running a fuel injected vehicle out of gas because they are hard to prime came from vehicles that didn't have pumps in the tank.
The pump in the tank is there to lift the fuel out, without a pump in there the high pressure pump must pull the fuel out over the top of the tank, and since it pumps liquids not air, it would have a hard time repriming, so don't run it dry
Fuel pumps run when engine runs, that is normal, so no worries.
Computer control of the fuel pump is about safety, if engine should stop running from an accident or roll over the computer cuts power to the fuel pump, when engine RPMs drop below 500, so you don't end up burning in a fuel pump fed fire.
Your slow RPM build up could be spark timing, or TPS(throttle position sensor)
Low fuel pressure, or dirty fuel filter, usually shows up at highway speeds first, so at 50mph it starts starving for fuel, but under 50 it is fine