Factual error: A few of Gruber's henchmen set up the rocket-launcher to blast the armored police vehicle. As one of them swings the legs of the launcher down, another uses a 'Hilti' gun to anchor the leg in place. A 'Hilti' gun is a construction device that uses gunpowder (usually a .22 shot) to 'blast' a nail into very hard material such as concrete. With the model they're using, the nail has to be manually loaded into the front of the gun for each shot before it can be used, however the guy using the gun never loads a nail into the gun. As it is, the result would be worthless.
Suggested correction: Hilti sells powder actuated fastening guns with collated fasteners and gunpowder shots lined up. Usually one can fire 10 rounds in a magazine. The gang would have probably selected such a model for this feature.
But that specific collated gun was not the one they were using.
Character mistake: An unloaded automatic pistol is much lighter and balanced differently than a loaded one. When McClane hands "Bill Clay" (Gruber) the empty pistol, a terrorist of Gruber's credentials should pick up on the odd balance (I notice that difference, and I very rarely even pick up a gun).
Suggested correction: This presumes that a person can notice the difference between a fully-loaded pistol versus a pistol that is empty, which makes sense. However, it would be much more difficult for a person to tell that a gun is completely empty versus one that only has one or two bullets loaded. Since Gruber has no way of knowing his ruse didn't work, he doesn't believe McClane would have given him a completely empty gun. He obviously believes there is at the very least one bullet in the gun. He's not planning on getting into a shootout with McClane, he is just planning to threaten him and he certainly doesn't need a fully loaded gun to do that.
Suggested correction: It's easy to tell the different between the weight of a loaded vs. unloaded pistol, but given Hans had never held the pistol before, he had no way to compare the weight. Unless he regularly handled that particular model of pistol (unlikely, given his pistol is a completely different make and model) he would have no point of reference.
Stupidity: Hans keeps a major part of his plan secret from his own team: that the electromagnetic lock will be disabled if the FBI shuts down power to the building. The mercenaries hired as muscle don't need to know the minutiae of the plan, but it seems ludicrous that Theo wasn't told. Theo states on more than one occasion that he can't proceed past a certain point and that he hopes Hans has a plan for the final lock. Evidently, Hans was keeping this information secret simply to amuse himself, which makes little sense considering how much planning went into the heist.
Suggested correction: Or because he simply doesn't trust anyone with that kind of knowledge. He neither trusts them or cares about them, it's all him.
So he trusts that Theo would be on board with all the murder and mayhem, open all the other locks, be in a tactical lookout position when the police try to breach, and drive the getaway vehicle. But he doesn't trust Theo enough to tell him the last lock will open when the power goes out?
It's not about trust; Hans needs Theo to do what he is there for and that is all you mention up to the final lock. He has a plan for the final lock and so there's no need to discuss it with the team, since it won't be any of them responsible.
The more people that know the plan the more chances of someone talking. Especially when they are hired mercenaries.
Theo was already on board with taking hostages and committing murder. Him knowing that the power needed to be shut off to open the last lock doesn't appear to be particularly important information you would need to keep from someone to keep them from talking.
If he's the only one that knows the final step to get the money, then at least up until that moment he is absolutely indispensable to the plan and ensures no-one would double-cross him. In any case I'm not sure being more cautious than necessary really qualifies as "stupidity."
Factual error: Would pulling the alarm call the fire and police departments if the phone lines were disabled?
Suggested correction: While I'm not sure how the alarms contact the fire department, it's also important to note that when Karl cuts the phone lines, his brother is re-routing some of the lines. I believe the implication is that he was bypassing certain lines so that way emergency services and the phone company wouldn't get alerts that their phones lines were down at the plaza and send people to check it out. So it would makes sense to me that a fire alarm could still get a signal through the phone lines in that case. (I'm also presuming that the way the fire alarm sends its signal is different from how a phone call is made, since one is automated and not call-based).