Fuel Pump in 2003 Neon SXT?

My sister's '03 Neon SXT (which was never properly taken care of by her... at one point she was 1.5 quarts low on oil. You could hear the problem about 50 feet away) has trouble starting. It'll turn over and crank for about 10 seconds or so, then start. The cranks are nice and strong, and the battery is fine. After that it runs fine, EXCEPT for when the pedal is floored in while it's idling. It'll stall out in that occurance.

We've been told that Neons have had fuel pump issues before, and I'm here to ask, would this be a fuel pump problem? The CEL is on (hope to get that checked out soon), and I just put in a fresh air filter to no avail. I've been told that it could be the injectors if she always ran on a low tank, but I would imagine it would run poorly throughout.

If it is the pump, is it possible to just repair the pump or send it in, or is a new/rebuilt one in order?

My Neon is a 1st generation (97) but some things dont change much. Your problem might be a fuel pump problem (low fuel pressure), but could also be a fuel filter or a bad fuel pressure regulator (I think these 2 are really one part), and probably other things that experts on the 2nd generation Neons will tell you, like Throttle Position Sensor or Manifold Air Pressure Sensor, etc. I wouldnt worry about the fuel pump until you had gotten the fuel pressure checked, and the CEL codes 'scanned'. My Neon has a Schrader valve on the fuel rail, and the fitting matches the a.c. gauges sets that were for R12 freon. So its easy for me to check my fuel pressure (49psi). I would hope they didnt eliminate those from the later Neons.
If the car runs all-right on the highway, and it accelerates well when you put the gas pedal down at 50 mph (that is it doesnt falter and cough when you do that), then its likely not a bad fuel pump. How long has it been since the car had new spark plugs and plug wires? You didnt say how many miles on the car. You might want to take the plugs out and see if they are a light tan in color, and not senile from the miles. Normally the factory plug wires will last longer than 5 years, but new plug wires are easier and cheaper to change than other things that wear out..
I agree with you that its not likely to be the injectors. If it runs on all cylinders at 30 mph, then the injectors must be okay.
The fuel pumps themselves dont get rebuilt - you would either buy a new one or a used one. Several companies make the pumps, so there different prices. Most of the time you wind up buying the entire pump assembly; the pump, inlet strainer, fuel level sender, and fuel filter/pressure regulator, and the in-tank housing. You can also get just the pump, and save money - but not time. And a service facility probably wont be interested in just replacing the pump. Its faster for them to replace the whole assembly and they can guarantee the result. Of course they cannot guarantee that was the only problem the car had.
If it is a fuel pump clogging up or going bad, dont wait too long - it may get bad enough that the car wont run at all. I would price out fuel pumps locally, and from places like www.rockauto.com and ebay.com. If you want to get the work done by a service facility or at a dodge dealership, expect the pump be pricey - maybe ver $400. To replace the pump you need to drop the tank; its a fair amount of work.

Thanks for all the info :). The plugs have never been replaced (going to do some sneak maintenence tonight to get new ones in... one cracked in half while I was checking it, pure luck that the thing's still sparking). It has about 81k miles, so I'd expect them to be getting pretty bad, especially when the OEMs are copper core, IIRC.

It didn't seem to sputter at high speeds. One thing I've noticed is that if it isn't run for awhile (trip to town), and turned off and back on again, it'll take awhile to start again. Would this be a bad filter/seal/something?

We'll be taking it to a local mechanic soon, as we lack an OBD II reader. Hopefully it won't be too bad.

Update: OBD2 code read as

P0340 - Camshaft Position (CMP) Sensor Circuit

Now what :P?