Stalling after high RPM rev

Those are both signs of a dirty throttle body and/or idle air control passages. The throttle bore on that car has a special coating to help prevent build up but it still happens. To clean it you need the specific throttle body cleaner that is safe for coated parts and a rag or long Q-tips. Run it a bit to warm it up. Now shut it off, remove the air intake duct and use the cleaner and tools to clean the interior of the throttle body. Next you will need to remove the IAC control. It can vary depending on which 2.0 you have on the DOHC engine it is below the throttle body and bolted on. The first two pictures show the location. The next three show the various types of valves used. They all bolt on the same way and act pretty much the same.
The last two images show the 2.0 SOHC valve. It works the same way. You just remove them, spray and wipe them out and reinstall.
Those valves set the idle speed of the engine, when they get plugged the idle will start to hunt because it has reached the limit of travel. Because of that and a dirty throttle body the throttle snap will close off the air into the engine and it will stall out. 90% of the time a thorough cleaning of the valves and the passages returns things to normal. Other times the valve has to be replaced because it has actually failed internally from the strain of the attempt to control it.

You may want to check the plugs for carbon. With partly plugged idle air passages it would have run slightly richer. The carbon can get hot and cause it to act that way.

Oil could be from valve stem seals, they are likely a bit worn with that mileage. You may also want to look at the PCV system. If the system has an issue it can draw in oil as well.