Stupidity: When Angela goes back to her house and finds that it's being sold, why are the cops letting her busybody neighbor hang around, ask her questions, and give opinions about her answers? If this were to happen for real, I think they'd ask the neighbor if she could identify her and then send her home.
Stupidity: In the scene with the water-jug puzzle, both Zeus Carver and John McClane initially suspect that the small carrying case contains a bomb. McClane goes ahead and opens the case, which confirms that it is a bomb (the electronic readout in the case even says: "I AM A BOMB. YOU HAVE JUST ARMED ME"). If they suspected it was a bomb in the first place, why didn't Carver and McClane immediately try to warn the dozens of pedestrians all around them to evacuate the area before McClane opened it? That would be a seasoned police officer's first instinct. Instead, McClane and Carver banter and bicker and never attempt to warn the public, even as the timer is ticking down.
Suggested correction: This carries over from the very first time McClane and Zeus talk to Simon on the phone (the "As I was going to St. Ives" scene). Simon tells the two not to run and McClane assures him they won't but that there are at least 100 people on the street, to which Simon responds "That's the point." McClane takes this to mean he can't just warn everyone or Simon will set the bomb off. They open the case because they don't know that doing so will arm the bomb. They wanted to know if there was a riddle or more instructions. Having the bomb be armed by being opened, complete with the pithy text "you have just armed me" was just a way for Simon to torment McClane. The bantering and bickering is of course, still pretty stupid but is consistent with how the characters behave for the entire movie.
Still, McClane's whole motivation in this movie is to save innocent people from death, which goes above and beyond his motivation in the first movie (which was to save his wife). What is the whole point of disarming the briefcase-bomb in the park if not to save the public? As stated in the original post, no seasoned and dedicated officer of the law would proceed without warning the public.
Except he was told specifically earlier not to do that. If he warned people of the bomb, it was implied that Simon would remotely detonate it. It can't be "stupid" of McClane to not warn people if he thinks doing so will get them killed. I agree that it is a trite movie cliche that a cop doesn't act like a cop would in the real world, but in the context of this film McClane's actions are consistent with the instructions Simon gives him.
