Fuel Problem

[InsaneCamaro]

I am having some major problems, when its cold out my car doesnt wanna start. When i open the door the dinger will soemtiems take a few seconds before it goes rright, then ill goto start the car and its very quickly dinging, but not constantly if it doesnt start dinging correctly it will jus shut off. My mechanic is a gm mechanic and he looked at my car i know it has an o2 sensor that is bad, but he told me its not the o2 sensor that would be doing this, his first idea was the fuel pressure regulator, he checked this and ruled it out because he said it would be leaking if it was going out. Sometimes when it starts it smells like its burning antifreeze, but there is not a drip missing out of the radiator or overflow bottle. I was also told the fuel pump would be going out, but then i think it wouldnt work at all , but once its started it goes great It has lost gas mileage. I was also told to look at the pass lock on the key and see if that was the problem, but i checked the ohmic value when i first got it and its still around the same value. O and so noone says this when i turn the key over i do let the pump prime and yes i can hear it prime. Sorry about the long azz post, but i wanted to get out all the different ideas i have and see what all of ya think and what different ideas you can give me. Im really hopping its not the fuel pump because there a dealer part only and they dont sell jus the fuel pump, but the whole assembly and its like three hundred bucks