Wouldn't be fuel pump, you notice that at highway speed when fuel demand is high.
Reads like IAC(idle air control) valve isn't working like it should, or you have a vacuum leak.
When you start engine cold idle should go up to 1,100 or higher RPMs and stay there, that's the IAC valve opening to let in more air.
Then as engine warms up idle will go down, IAC valve closing bit by bit over next 5 to 10 minutes.
Once engine is warmed up and idling, unplug the wires on the IAC Valve, idle should drop to 500 or engine may stall, either is good it means no vacuum leak, idle 700 or higher means vacuum leak.
The computer controls the IAC Valve
On a fuel injected engine there is no "gas" pedal, like on a carb engine, when you press down on that pedal now you are letting in more air, not adding gas, so it is an "air" pedal, lol.
Any way your description of having to press down on the "gas" pedal when starting could mean IAC Valve is not opening all the way for starting.
When you turn on the key the computer opens IAC Valve all the way to get in enough air to start engine, if that doesn't happen then using the gas pedal would do the same thing.
You can remove the IAC Valve and turn on the key, you should see it open all the way, then unplug it's wires and it should close all the way, repeat a few times to make sure it isn't sticking.
Long shot is electrical system, do the head lights dim at idle?
If so then alternator could be failing and drawing voltage down at idle, which reduces spark voltage so misfires and stalling when idling, but you should begin to notice slow starter motor cranking after a few days of that.