Continuity mistake: When the villains arrive at the Daily Planet, the furniture, decor, and buildings in the background change between shots.
Other mistake: When Lois and Clark leave the Fortress, the car is driving in the wrong lane. This is due to the fact that the shot belongs to a reused scene from the first movie, where an empty car was remotely driven by Lex, who changed lanes all the time.
Factual error: When the Krypton criminals are on the moon, they speak both to each other and the astronauts, something that's impossible to do in space where there's no sound.
Continuity mistake: After the Metropolis fight, the villains return to Perry's office. In the close-up, the whole background changes: the buildings, the debris around, and even a column and a broken blind appear behind Non.
Revealing mistake: All of the movements of both people and objects on the moon are played in slower speed to simulate the Moon's gravity, except for the last scene where Zod grabs the oxygen tube and it falls on the ground at normal speed.
Continuity mistake: Before Non breaks the table in two, the phones are lying next to the edge. When he raises his fist, the phones are further away.
Audio problem: In the prison, Lex tells an inmate he wants his Liberace record, to which he replies, "It has a scratch on it," but his lips don't match the audio. In the Donner version, it's revealed that the original audio was "Tonight?"
Continuity mistake: At the Fortress, when Lois drinks champagne, she holds the glass at chest level. A shot later, her arm is lowered.
Revealing mistake: When Non flies through the White House ceiling, right before he lands on his feet, the shot cuts to Zod landing safely and one can see that the previous stuntman missed the landing and fell on his back, his legs now pointing upwards. According to the shooting script and the additional Donner scenes, the fall was not scripted and not plot-wise.
Revealing mistake: When Superman returns the flag, the wide shots show he has really long arms and a brown blob where the head should be, revealing the actor has been replaced with a dummy.
Continuity mistake: Right before Perry White throws a bronze globe at Non, Lois' head is tilted sideways and she smiles calmly. An instant cut later, her head is straight and she stares in awe. After the globe strikes Non's head, we're back to the previous shot where her head is tilted.
Visible crew/equipment: In the town in Idaho when General Zod and his crew first visit, watch when the boy's father (elevated by Zod's finger) falls to the ground. You can see the pad under the dirt when he hits it.
Visible crew/equipment: Ursa throws the sewer lid at Superman sending him flying backwards into the car windshield. Watch the sewer lid carefully when Superman actually hits the windshield. You can see that the sewer lid was actually attached to a belt around his waist so that it would stay in place for the shots where he is flying backward through the air. (01:39:50)
Revealing mistake: In the final scene, Superman is returning the White House's flag to its rightful place, flying over the fountain on the lawn...which is frozen solid in place. Obviously a photo backdrop.
Continuity mistake: When Superman starts playing hide and seek with the villains, he stands away from the icicles. When Ursa shows up a frame later, he's leaning against them.
Revealing mistake: When Zod lifts the boy's father with the beam of white energy, watch closely when he raises him up past the telephone pole...he 'clips' right through it, or rather the pole clips through him, as if he is transparent.
Visible crew/equipment: When Superman is fighting Non under the New York Streets (in the sewers), manhole covers start to jet up in the sky. When the first manhole cover shoots up you can quite clearly see the gas tube in the hole and the gas cylinder attached to the cover.
Revealing mistake: There are two external White House shots in the film. The second is a still shot, and should take place at least a few days after the first. However, it is a frame from the earlier shot, with the same person standing in the same spot on the lawn.
Factual error: After Superman accidentally frees the phantom zone villains with the exploding hydrogen bomb, they start flying towards the moon. The sleeves in their clothes can be seen fluttering around. This would be impossible because there is no air in outer space, hence nothing to cause resistance against the clothes.
Other mistake: When Clark and Lois are on their way to the diner, the wide shot of their car driving along looked very familiar. In fact, it was footage used in the first movie, where the car was used as part of Lex Luthor's "real-estate" plan and was run on remote control. If you look closely in part II, no one is driving the car yet you hear Clark say to Lois, "Hey, you seem pretty quiet over there...."
Answer: Superman was be able to kiss Lois in such a way that the heightened emotion wiped out her memory of his identity. To answer your second question, even though Lois promised not to tell anyone, Superman decided he does not want even her to know his secret identity.
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