Stupidity: Oldboy finds the right Blue Dragon restaurant that the prison uses. But him running after the delivery boy doesn't make sense. How does he know the delivery boy is going to the prison? He knows it's the right takeaway, but that's all he knows. So, would he have a moped, bicycle, or car ready to use surely? The takeaway is a long way from the prison (as he says himself in narration), so he wouldn't just wait around for a moped delivery to run after, especially as a good chance he wouldn't be able to keep up.
Stupidity: Why would Stanley say the shoes fell out of the sky? It happened under a bridge. Any normal person would instantly realise someone threw them off the bridge.
Suggested correction: It's an idiom/expression. When something turns up out of nowhere, we say "it came/fell from the sky." In fact, Mr. Sir says this to Stanley when he finds Stanley with his empty bag of sunflower seeds, "Where did this come from? Did it fall from the sky?" Obviously a sarcastic way of saying, "it didn't get there by itself, someone put it there." So the shoes obviously fell from the bridge and Stanley knows this because Zero refers to the bridge later in the movie.
Stupidity: If the doctor was able to choose the people to go on the helicopter, why didn't she make up a reason to put the president's son on the helicopter for his safety? She could have said he was sick. (01:14:40)
Suggested correction: He is shown to be a caring person. There is no way he would take the place of a sick woman or child just to save his own neck.
That correction is illogical though. By not getting on the helicopter he puts everyone else at way more risk. It would have made getting the rest of the people left behind out of danger much easier as they wouldn't have been hunted down by the people who were chasing the president's son. By not getting on that helicopter he could have gotten everyone killed.