Revealing mistake: In several scenes Superman has fillings in his teeth.
Revealing mistake: When Jor-El is placing the baby Kal-El into the spacecraft, you can see his Rolex under his sleeve.
Revealing mistake: When Lois is at the heliport, watch the wind flag on the left: when the helicopter arrives, the roaring wind from the vanes makes the flag move up and get stiff. In the close-ups, you keep on hearing the sound but the vanes are off screen. Watch and you'll see the wind flag is down, thus revealing that the engine is off and that the sound we are hearing is fake.
Revealing mistake: When Superman holds the plane's engine, the "S" on his shirt is looking the wrong way, revealing the angle was flipped.
Revealing mistake: In the shot of Krypton exploding, you can see creases in the black backdrop in the lower, left hand quadrant of the screen (especially visible on Blu-Ray).
Revealing mistake: In the final shot of the destruction of Krypton several of the people falling into the abyss disappear a split second before the shot of Krypton exploding.
Revealing mistake: Superman startles a cat burglar who falls from the side of a building. When Superman catches him he is facing upwards yet his cape does not fall back, it stays parallel to his body.
Revealing mistake: When in the prison yard, Luthor's bald cap is hanging loose around his nape.
Revealing mistake: When Lois falls from the helicopter, the cable holding her is visible.
Revealing mistake: After saving Jimmy from falling to the dam, during the flying sequence a couple of wires are seen between them.
Revealing mistake: In the extended version, when Luthor fires snow at Superman, the icicles bounce, rather than shatter, revealing they are some type of foam or plastic balls.
Revealing mistake: When Otis first arrives in Luthor's lair beneath the train station, he knocks a table lamp over. When the lamp shade comes off the lamp, the bright, exposed light bulb then dims to a less-distracting level for filming purposes. Otis then replaces the shade upon the lamp, and the bulb then instantly goes back to its original level of brightness, so that it will look appropriately illuminated through the shade. The bulb's brightness was manipulated off-screen to conform with the needs of the bulb's effect on the scene.
Revealing mistake: When the helicopter gets jammed on the white railing at the ceiling, Lois is replaced by an obvious dummy that dangles back and forth with no control.
Revealing mistake: As Air Force One enters the frame from the right, there appears to be a set of tracks faintly visible under it.
Revealing mistake: When the helicopter is hanging off the roof, a safety securing cable is attached to the fuselage to keep it in place.
Revealing mistake: On the Golden Gate bridge, the beetle car that skidded a split second before appears totally empty when the van crashes against it.
Revealing mistake: After jumping in front of the train, Clark runs leaving a white trace behind. Watch the grass on the lower right and you'll see the scene is speeded up.
Revealing mistake: Look carefully at the train when Superman is using himself to hold up the track during the earthquake. It's obvious the wheels of the train are not even touching the track as they pass over him.
Revealing mistake: When Superman and Lois begin their flight from her apartment, a pair of wires suspending each of them are visible, especially on the Blu-ray.
Revealing mistake: All the footage of the missiles being launched is of a poor quality and grainy, which is inconsistent with the good quality of the rest of the shots, revealing that it's stock footage.
Answer: He wanted to see for himself if the stories were true. Some reporters tend to exaggerate the facts and if he had any other weaknesses. He couldn't be sure the kryptonite would work.
While I don't exactly disagree that Lex wanted to see for himself how invincible Superman is, I don't think that's the main reason why he did it. The bullets and fire were simply a charade to make it look like this was his security system. Keep in mind, he wanted Superman to enter his lair because the real trap was the Kryptonite that he had in the lead box.
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