Q: Hard start, sputtering even idle, sometimes turns off while idle

My car has a hard start. After resting the night, I start it in the morning and usually get the red oil light in the dash. If I turn the car off and start it again, the light will not come on. (this gives me a little peace!) while not moving, it will start to shake or sputter. Car improves drastically when I start driving. Sometimes I have a little bit of difficulty accelerating, but if I get on the highway everything is okay. Once I have to stop at traffic lights or traffic jams, the sputtering begins again and in some cases it will turn off. (it has always started again. . Thank God!) also, the rpm needle seems to have a life of its own. When stopped and shaking, the car seems to be trying to self adjust and the rpms go up and down. While doing this the lights seem to flicker a little bit. I have gotten different diagnosis.. A lot said fuel pump. When I look for this part online, I can get just the pump, or the pump assembly which includes fuel sender. Big price difference!

My car has 133000 miles.
My car has an automatic transmission.