Car will not stay started?

The clue is the battery. Before we go any further, go out and drive the vehicle at highway speed. Once the engine is warmed up, coast for at least seven seconds without touching the pedals. That will meet the requirements for "minimum throttle" to be relearned after the battery was disconnected. Once that is relearned, the Engine Computer will know when it must be in control of idle speed. Let me know if that solves the low idle speed.

Dandy. The next step requires a scanner so this might be a dealer item. I found two procedures related to idle speed, shown below. This is a throttle-by-wire system, so there may be no relearn procedure like I described. Instead, look at these articles first.

Next, have you checked for any diagnostic fault codes? If you find code P0102 is set, "Mass air flow low", check the operation of the intake air temperature sensor. A broken wire or failed temperature sensor will cause low idle speed. The last diagram shows that sensor with arrows pointing to the two wires to check. It is necessary for the sensor to be plugged in for the readings to be valid. Back-probe through the rubber weather seals alongside the two wires to take the readings.

You should find less than 5.0 volts on terminal # 1, (red arrow). 0 volts and 5.0 volts point to a defective condition. Typical is between roughly 1.0 volt to 4.0 volts, depending on air temperature. If that is right, the second wire will be right too. That one, terminal # 4, with the blue arrow, should be 0 volts, but expect to find close to 0.2 volts. The point is we don't want to see anything higher than that.