If its very cold or if its raining heavy or its a very humid day, when I start the car occasionally for 10-30 seconds the car has severe difficulty getting to idle and stutters/sputters severely. Repeatedly pressing the gas lightly seems to help get some gas to the engine, but completely depressing the gas pedal doesn't seem to help any more than a light tapping.

Eventually it corrects itself and goes to idle without difficulty, but twice now I've been unable to actually start the car after 5-10 minutes. And once I ran the battery dead trying to get the car started. The engine turns over fine, but there definitely seems to be a problem with gas getting to the engine in these very cold, or very humid/raining conditions.

I've replaced the fuel filter because it was something I could do and was cheap, but no improvement.

Does this sound more like a sensor problem (o2 or MAF?) or a fuel pump going bad? If the fuel pump was bad would it show signs like this only when starting up or wouldn't it be a continual problem? Something wrong with the fuel injector?

It sounds like something I need to take in or have codes run on to get a better idea of whats going on. But figured I would get a better idea from more experienced people and know what all the potential possibilities could be.

 

MAF and O2 sensor do not operate for the first ten seconds or so, so we can rule those out.
Moderate fuel enrichment is normal when starting up, the wetter the worse it can be. Sounds to me like the enrichment is too rich, so here is what you should do to cut the places to check in half.
Try and start it for ten seconds or so, then stop and pull a spark plug. Is it wet, dry? Wet, too much fuel, possible ECT failure or beginning to fail. Dry, lack of fuel, possible fuel pump relay going bad, or the fuel lines are purging when you are trying to start it, bad check valves...

You yourself can check the codes with the OBD, do that and it could take you to your source right away.