Smoking from valve cover area?

Those are not good signs. It is very likely that the overheat caused the head gasket to fail, that would lead to the issues you have. The 2.5 isn't a good engine to overheat as the heads like to warp. I would pull the plugs and see if you find wet plugs, if so then either a remanufactured head (if the block measures out flat) or a complete engine if the block shifted, would take care of the problem.

I agree with STEVE W. here are tow guides that will help confirm the head gasket is blown due to overheating.

https://www.2carpros.com/articles/head-gasket-blown-test

and

https://www.2carpros.com/articles/recognizing-the-symptoms-understanding-a-blown-head-gasket-in-your-vehicle

After confirmation you will need to remove the cylinder head and send it out for a crack test and resurfacing, here are the torque specs and sequence. Please upload a short video in your response of the problem, so I can see what's going on :) Check out the images (below). Let us know how it goes

How long was that after start-up? Take a sniff of the smoke, is it sickly sweet like burnt sugar or rancid smelling like burnt tar? First one is coolant second is oil. It may have gotten hot enough to fry the valve cover gasket and dump out some oil. I would wipe down the area and dust it with some flour or similar, then start it. Watch the flour. Does it turn dark like the oil or just get wet like coolant? If it's dark it could be oil, however that wouldn't make it run like you say it is. Stalling and stuttering would still point to gasket damage.

Hmm, that looks like oil burning to me, was the engine hot? You could have a valve cover leak. Did you try the chemical test to see if you have blown head gasket?

That engine has a single O2 sensor located in the exhaust pipe just below where it is attached to the exhaust manifold. They can be a bear to remove. Generally I like to run the engine, then let it cool down and as it cools, spray some penetrant on the area where the sensor screws into the pipe. Then once it's cold, remove the sensor.

You can try it but I'm thinking it has a bigger issue. Might want to pull the plugs and do a compression test, then if you have compressed air available connect it to each cylinder in turn and slowly apply air pressure in the cylinder with it at TDC on compression. While you do that look at the coolant with the cap off. If you see bubbles or worse the water starts rising or "boiling" from the air you know there is a problem.