Car has problem starting

for a few months now when i go to start my 2002 GT it turns over for like 5-8 seconds and then sputters a little bit and then starts for a little bit and then dies. If you try and start it again it does the same thing and then starts and stays running. It will stay running the first time if i give it gas once its started. It will start right up after the first turn if its really cold out. I thought it was the O2 sensor at first but it isnt throwing any codes. Anyone have any ideas?

How does the pump sound when you turn the key on but don't start it?

Both when it's cold and starts up and after it's warm.

i removed the vaccuum line from the fuel pressure regulator and there was fuel in it, so i replaced it and now everything is working fine

Crankshaft position sensor???????