Stupidity: When Spider-Man confronts Aleksei Sytsevich in the truck while he's plowing through the busy streets of New York, he sits there and starts cracking jokes instead of immediately trying to stop the truck, effectively allowing Sytsevich to potentially injure or kill numerous people in the process, as well as to wreck many vehicles. He even allows Sytsevich to fire a gun out of his window and into the open crowd instead of trying to disarm him as soon as he sees the gun.
Stupidity: Peter Parker watches Richard Parker's last message, the upload he did on the plane to his super encrypted server. As he does that, the interface used is one of a mail and there's even the sender; "Rich-Park@oscorp.web." So Richard Parker sent his dying message he wanted to keep from the evil corporation...through his evil corporate mail account, and to an account that contains the location of his secret lab, "Roosevelt." (01:32:50)
Stupidity: Menken runs away from the Special projects section inputting an alarm code in there. Security at Oscorp is not exactly well-thought-out though, because apparently when there's an alarm in their secret weapon-experiment containment facility, their security protocol consists in opening the cells containing those super secret projects so any intruder can more readily steal them.
Stupidity: Harry Osborn quite literally inherited the company his father founded. Presumably he owns or controls a majority of the stocks, because he was appointed CEO by his father and nobody questioned him. However, one of his employees can just instantaneously fire him from his position. We don't know the precise rules and internal regulations of Oscorp, but it's safe to say that this is not how company hierarchy works, especially considered that no charges are pressed on Harry and everyone would be out of a job (including Menken) if the circumstances were public - like having created a monster and waterboarding a guy in their basement.
Stupidity: Oscorp wants to keep hush-hush the death of Max Dillon and erased entirely his existence from the records, because "he is invisible", and a big plot point is made that the guy is 'invisible'. Let's go by this premise, even if it's patently absurd (the guy is so conspicuous that he's going to be the butt of plenty jokes, and we saw at least two coworkers he's familiar with); erasing the records and pretend he never was an employee makes sense, but actively hunting down Gwen Stacy just because she looked his name up -finding nothing - does not. Imagine if Peter didn't save her and they caught her; there's nothing Oscorp security could have done to her without exposing them much worse.
Stupidity: The security fails to block Gwen Stacy at floor 63 of the Oscorp tower. They own the whole building, and they have radios; by the time she reaches the ground floor (which is exactly the floor we see her reach) the building security can intercept her. We saw in the first movie (and it's basic logic and logistic of a company of this size) that they do have security at the entrance - and it's the regular building security that is involved, not a couple of mercenaries.
Stupidity: In the closet with Peter, Gwen says that "there was an accident in the genomics lab, and they're covering it up." That's a pretty exceptional piece of information that she has, but this bit of information is never explained, investigated, deserves the slightest attention, just because the plot says so. Peter does not even ask "what kind of accident, what happened" or anything. (01:09:40)
Stupidity: Crowds in movies are not renowned for their brilliance, but the crowds in this movie are formed by idiots. Both in the Electro and the Rhino fight, pedestrians are standing by behind strategically placed railings (that have no reason to be there to begin with), just to witness first-hand from a few meters the lethal powers of rampaging monsters - and they keep their little kids with them too.
Stupidity: The evil corporation (and the FBI too) wants to use Richard Parker's work against his will, but he knows that without him they can't replicate the experiments. So what does Richard do? But of course he escapes making sure to leave his son behind; if the evil corporation didn't kill Richard, they could have easily blackmailed him through his son. That's of course assuming they would never figure out on their own that his DNA is compatible and use it - a risk Richard knows given his choice of words mentioning "my bloodline."
Stupidity: Richard Parker didn't have an escape plan for his family or for erasing his experiments, but on the other hand he had a custom James Bond-like secret laboratory hidden in an elevator of a metro station - something he surely couldn't have built in his spare time, and yet at the same time he seems to be 100% sure that his boss does not know about it because he sent to it a pointless message that gives away the secret he gave his life to protect. The lab has a private satellite feed and stayed powered for the whole 12 years of his absence, too.
Stupidity: Spidey at the beginning of the movie strips out of the costume in literally a second, but when Aunt May is at the door of his room he fumbles around for an eternity and is unable to change just for the sake of a gag. This pairs up with another nonsensical gag; Peter is in a convenience store, dressed normally. In a split moment he wears the costume, and he put the jacket, hat, scarf, backpack on top of it. If the store had a camera, or he uses those clothes normally as Peter, his secret identity is compromised.
Stupidity: Richard Parker needs to upload the video message with his 'confession'. The entire operation requires him mere seconds but instead of taking those extra seconds to do it from home (the bad guys already ransacked his place), he waits to do it when he's on a plane, which in 2002 was not really a great option. It should also be noted that he says he is uploading "all of it", but the only thing uploaded according to the computer is the video.