Q: Car idles and dies on cold start but starts when warm

It sounds like you may have a fuel supply problem in getting the proper amount of fuel to the fuel injectors. The fact that the condition gets better when using starting fluid also points toward a fuel supply problem. As you know, when you turn the key to on, the fuel pump is activated and the fuel system is primed with fuel at a specific pressure in order to supply the fuel injectors with enough fuel to start the motor. When this is less than what is needed in order for the motor to fire or in some cases higher than normal, you may notice an erratic fluctuation in engine RPMs, hard starting or a lack of power when accelerating as the engine may be starved for fuel if insufficient fuel pressure exists in the fuel system. This can happen when the pump itself is not working properly or when the fuel pressure regulator is not maintaining the proper fuel pressure in the fuel system causing the engine to starve for fuel. This can also happen when there is insufficient connectivity to the fuel injectors which can cause an intermittent fuel spray pattern into the intake system resulting in a lean fuel condition or the motor otherwise starving for fuel. The other possibility may be a bad coolant temperature sensor which relays the temperature of the coolant to the computer. During cold starts, if this is not working properly, this may cause inaccurate fuel adjustments to be made resulting in the rough idle problem. I would recommend having an expert from YourMechanic come to your location to diagnose your fuel system.