Why won't it stay started

Need a whole lot more detail than that. There's a dozen different symptoms that your description could apply to.

Well, there's good news and less-than-good news. First of all, the fault codes did not say to replace any parts. They never do. They only indicate the circuit or system that needs further diagnosis. Indeed the sensor referenced by the code is defective about half of the time, but the other half there is some other cause like corroded connector terminals, a cut wire, or in the case of a coolant temperature sensor, something as minor as low coolant level. There are no codes for the thermostat. The only code related to that is "running cold too long", and every Chrysler product has that one in winter. It sets when the engine doesn't get up to proper temperature within six minutes of starting it. That code will not turn on the Check Engine light and is generally ignored.

There are two coolant temperature sensors. One has one wire and it is for the dash gauge. The other one has two wires and is for the Engine Computer. If you unplug that one while the engine is running, the radiator fan will turn on. That is a fast way to verify the fan motor and relay are working and the computer has control of them. It does that because with that sensor unplugged, there's no way for the computer to know if the engine is overheating, so it turns the fan on just in case. Unplugging that sensor while the ignition switch is on will set a fault code and cause the Check Engine light to turn on. On some engines the light will turn off as soon as you reconnect the sensor. On others you have to turn the ignition switch off and back on, then the light will be off. Either way the fault code will stay in memory, but it will erase automatically after 50 engine starts.

There are about 30 fault codes related to the oxygen sensors so we have to know the exact number of that code to know why the computer is unhappy. Only about a third of those codes are actually caused by the sensors.

The new battery terminal is the reason for the engine stalling and it's why I asked about that recent history. Disconnecting the battery cable made the Engine Computer lose its memory. Short and long-term fuel trim data will start to be rebuilt as soon as you drive the car, but it takes specific conditions for the computer to know when to put "minimum throttle" in memory. Until it learns that, it won't know when it has to be in control of idle speed, and stalling at stop signs is the common result. You also won't get the nice "idle flare-up" to 1500 rpm when you start the engine. In fact, it's common to need to press the gas pedal 1/4" to get the engine to start. To meet those conditions for the relearn to take place, drive at highway speed with the engine warmed up, then coast for at least seven seconds without touching the pedals.

White smoke from the tail pipe is a sign of burning engine coolant but that can be hard to tell in winter. Keep an eye on the level in the reservoir. It will go up when the engine is warmed up, and it will go down as the engine cools down, but those levels should remain fairly constant. If the level drops over time, there are two tests your mechanic can do to verify a leaking had gasket is responsible. One involves drawing air from the radiator through a glass cylinder with two chambers partially-filled with a special dark blue liquid. If the head gasket is allowing combustion gases to get pushed into the cooling system, that liquid will turn bright yellow.

He can also add a small bottle of dark purple dye to the coolant. Once the level drops in the reservoir, he will search with a black light. The dye will show up as a bright yellow stain that can be followed back to the source of the leak. With a leaking head gasket, the dye will show up inside the tail pipe.

A leaking head gasket can allow combustion gases to pool under the thermostat and prevent it from opening. They open in response to hot liquid, not hot air. That can also prevent the temperature sensors from working properly, and that can prevent the radiator fan from turning on. The resulting overheating can cause the cylinder head to warp and leak. What that means is a leaking head gasket can cause overheating, and overheating can cause a leaking head gasket. It's often difficult to determine which one caused the other, but for sure, if the fan isn't working, it is likely to lead to more expensive repairs.