Well, my theory is that at that age and milage, many parts start to not perform anymore as they should and the combination of little variations makes for the annoying and hard to find small problems. If the injector seals are leaking a bit (may be more on a cold engine) it might take the warming up period to make them seal better. May be the idle control valve needs cleaning. You can look at the online catalog from FastLane how many parts are involved in fuel injection and fuel delivery.
There is a test for the idle control valve. When pulling off the electric connector, the idle is supposed to go up to 2,000 rpms or so. There is also a lot in previous posts how to test the O2 sensor with a Multimeter, but I don't think in your case the O2 is at fault, as it seems to be working once warm. Cold start valve seems also ok.
Spraying carb cleaner at the injectors might reveal a leak there, but to detect it better, the exhaust emission needs to be read simultaniously.
Did you try injector cleaner? That would be my first action.
BTW, I prefer driving the car right away with low rpms rather than let it rev for 2 minutes at 2000 rpms.