Other mistake: When John and Zeus are being chased by the German guys in the car, John asks Zeus to pull out all the fuses in the fuse-box. The next shot is a quick shot of the speedometer etc. and it is lit up. Surely if all the fuses were out it wouldn't be lit?
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.
Visible crew/equipment: When Zeus and John get the cab after being attacked by the group of dudes in Harlem, as they are riding to the police station you can see the camera truck out the back window of the cab.
Suggested correction: How do you know that's a camera truck? Could be just any regular truck passing by.
There is a camera held out of the passenger side window of the van.
Character mistake: Why didn't Joe, Connie, and the students just run back down to the front entrance of the school? They had plenty of time instead of making the effort of going to the roof.
Suggested correction: No, they couldn't. They barely made it to the roof on time and that was closer than the front entrance.
Suggested correction: No, the car would interpret missing fuses as blown fuses and would display a light to highlight the error.
GalahadFairlight
In German cars, the lid of the fuse box shows a table listing the purpose of each fuse. The engine control unit, fuel pump and the dashboard also have fuses that you should avoid to pull.