Other mistake: While the fire on the oil rig burns all around, Mr. Freeze grabs Belson, then orders him to go find Barbara (to do the transplant). When it cuts to a closeup of Freeze, the reflections of Belson on Freeze's eyeglass lenses are not reversed as they should be. (00:53:00)
Continuity mistake: On the burning oil rig, when Robin and Barbara pull Mr. Freeze out from underneath the rubble, Barbara is missing her shoe and blue sock in this shot. (00:54:20)
Continuity mistake: Belson releases the mooring line to take off with the boat, and we see the rope slip away in close-up. Too bad that in the shot that follows the ignition, the boat is still moored, or at least the line is still in place. (00:55:35)
Continuity mistake: Batman is hit by a falling piece of debris as he tries to save Freeze. He seemingly lets just go a button (or something) of the bat-rope gun, since it's still in hand, but a moment later he is dramatically reaching down calling Freeze's name and the hand is empty. He then stands, bat-rope in hand. (00:59:05)
Plot hole: Batman finds at Belson's home a handwritten list of substances needed for a transplant. Why would Belson, a medical authority in the field that knows those things thoroughly, even write a list down? If it's for Freeze to procure him the material, why would it be at his house and not with him, and when did he even have time to be at home to do that, since he has been kidnapped by Freeze when he was driving on the outskirts of Gotham and taken to the oil rig? If he had time to go home, he at least would have changed clothes, he's still in the tuxedo he wore when he was kidnapped the night before.
Continuity mistake: The heels on Barbara's shoes appear and disappear throughout the movie.
Plot hole: There are 17 possible donors for Mrs. Fries. Of all of them, instantly and with no explanation, the doctor chooses Barbara Gordon. She was not the first alphabetically, from the phone call they find out she has a roommate, a boyfriend, she is out for dinner at the moment and is going out of town and of course, REALLY huge warning sign, she is the daughter of the police commissioner (hard to ignore that in Gotham, and Freeze certainly knows him first-hand!). There's no reason why to kidnap her, plenty of reasons not to, and zero reasons given why they did not try any of the other 16 names on the list.
Suggested correction: They did explain exactly why they picked Barbara. The names on the list weren't all perfect matches. They were looking for someone "who's approximately the same weight, height, and female." When he sees Barbara's file, she's a perfect match. The fact that it's Jim Gordon's daughter is of no concern to them, assuming they know it's his daughter since Gotham is said to have a population of 10 million.
You are quoting the movie but then describing something radically different from what is shown (which I understand, since what is shown makes little sense). The list of 18 people (one being Mr. Freeze's wife) is the one that appears exactly *after* he has refined it for "approximately the same weight, height, and female." Everyone on that list should be a perfect match, otherwise there was no point at all in showing him writing F, 5'2", 116 lbs, AB Negative in the search fields and get search results for it. If it's not, he has no way to tell at first glance.
Other mistake: Freeze uses his gun to extinguish the fires on top of the elevator, and that somehow gets it back to work, but it does not really make sense. His gun greatly weakens metal, as shown earlier, to the point of breaking it, and he did not shoot the engine of the elevator, which is what was shown malfunctioning.
Factual error: The plot revolves on finding a matching organ compatible with Nora. Practically the only issue presented about it is the blood type compatibility, so let's go by that logic. In a misconception happening in so many works of fiction, the fact that she is an AB - is listed as a terrible problem because of how rare it is. It is indeed the rarest blood type in most parts of the world, but it's also the universal receiver and compatible with every other group.
Continuity mistake: During the fundraiser, glasses of relevant guests (the woman in the red dress when she is introduced to Bruce, the stock broker when he is dragged away) disappear conveniently off-camera. It's quite a stretch to assume they found a waiter or a place to place the glass when they were not in frame (especially in the group of three women in front of the doors).
Other mistake: Freeze's hideout is in an abandoned oil rig, with its tanks full to the brim of oil. You'd think the company owning it would have drained the millions of dollars' worth of content first, before 'abandoning' it.
