Continuity mistake: After being hit by a cab, Lois calls Clark and the people behind her change in a split second.
Continuity mistake: In the big fight scene in Metropolis, after Ursa chokes Superman, Non descends and wires holding him is visible.
Continuity mistake: When Superman is about to stop the elevator, he flies with one arm up the other down. A frame later in a new angle both arms are up.
Continuity mistake: On Krypton, when the guard gets caught, Non's hand is at chest level. In the close-up it is at belly level.
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.
Other mistake: When the three villains jump out of the Daily Planet window they just break some glass; however, from a ground level shot huge chunks of concrete are falling. This is more dramatic, indeed, but senseless.
Revealing mistake: During the Metropolis battle, after Non and Super go down through the pavement, Non is launched upwards thorough the concrete. Check the people around, specially the cop on the right. They are blatant toys (note the cop's weird arms), revealing it is all a miniature set.
Audio problem: When Non is launched towards the Empire State a man says "Wow, home run!", but his lips are not in sync.
Revealing mistake: When Non punches Superman and he crashes against a metal wall, there's a brief shot where Chris Reeve's stunt double is very noticeable.
Continuity mistake: Inside the White House the cops fire 3 or 4 times at Zod but the wall and statue behind him have at least 15 bullet holes.
Other mistake: When the elevator explodes there's a shot of the Phantom Zone and Ursa is a totally different actress. Different to the other unknown actress used on the first movie, which is also listed as a mistake.
Continuity mistake: Right after Clark reveals himself to Lois in Niagara Falls hotel room, by his hand falling into the fire, he backs up to where the bed and lava lamp like column is behind him. In that shot, there isn't much of the "lava" in the column. The scene cuts to a shot of Lois and then back to Clark and suddenly there is a lot of "lava" in the column. (00:53:05)
Continuity mistake: In the diner, whilst Clark is asking the bad guy to step outside, the waiter is leaning over the baddie's right shoulder. When the angle changes he is leaning on the left shoulder.
Continuity mistake: When the villains are being judged, for a brief second they change positions, Zod being on the far right instead the far left.
Audio problem: At the Eiffel Tower, after Lois asks the French officer if the terrorists have the proper equipment, his voice is heard but from a side view his mouth and jaw don't move.
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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