Continuity mistake: Note the chrome centers on the wire wheels of the taxi that Jack commandeers towards the end of the film. Initially the left rear one is missing, but it magically reappears later on.
Continuity mistake: Soon after Kong grabs Ann for the first time and is playing with her the wide shots show him violently shaking her around, enough to snap her neck or at least render her unconscious, but the close-up shots during the shaking show her being much more gently rocked from side to side.
Continuity mistake: As Jack Driscoll flees Kong in a yellow cab through a narrow alley, the view through the (missing) windshield shows a pedestrian in light-color overcoat crossing leisurely at the other end. When the car eventually emerges in the immediate next shot viewed from the sidewalk, that particular pedestrian simply disappears.
Continuity mistake: At the diner, Carl speaks with Ann while she eats, and things on the table change position depending on camera angle.
Continuity mistake: When the crewman receives the message aboard the Venture, he jots it down on a pad of paper. There is nothing under the pad in one shot, but in the previous and following shots there are more papers under the pad.
Continuity mistake: As Ann tries to escape from the T-Rex, at one point her robe acquires a rather large black and muddy stain at her backside on her left, yet in the very next shot that large black stain is gone.
Continuity mistake: When Carl offers the Nestles chocolate to the native girl, the wrapping differs between shots as he holds it out to her.
Factual error: In at least one shot as Kong is on top of the Empire State building, the sunrise over Long Island is wrong. Three planes attack from the southwest and we see the sun rise in the east as Newtown Creek (Nassau River) glistens across the East River. The mouth of this river opens southward but in this shot everything is inverted with the mouth of the river pointing northward. Also, looking eastward, Roosevelt Island should be on the left of Newtown Creek, not on the right.
Continuity mistake: On the cliff where Ann is entertaining King Kong by dancing, juggling, doing flips, etc. her face goes from dirty, to clean, to dirty, and back to clean between shots.
Factual error: When Denham sees the smashed camera he sees the film stock spilled on the ground. It is brown, indicating unexposed colour stock. Brown emulsion film didn't exist in those days - it was introduced by Kodak in 1943. In those days it would have been black and white emulsion, blue on one side, and grey on the other. Colour films in the early 30s were made by painting black and white films frame by frame or printing the monochrome negative three times and staining each one a different primary colour.
Continuity mistake: At the end of the chase through the streets of NYC, Kong stops the yellow cab with a "karate chop" to the front-end, which sends the car tumbling high into the air through a complete somersault before it lands back down. Actually, Kong's hand is still half way down and nowhere near touching any portion of the vehicle when its rear-end gets lifted.
Factual error: There is no lift in the Empire State Building running between 1st and 100th, as shown on the dial in the lift.
Continuity mistake: When Jack is rescuing Ann she has two cuts above her right eyebrow, the smaller cut changes positions and disappears/reappears in different shots.
Factual error: The drop fare on the taxi is shown as 25 cents. However, this fare was not charged until 1952, when the former taxi fare of 20c 1st 1/4 of a mile and 5c each additional 1/4 of a mile (which went into effect in 1934) was increased to 25c the 1st 1/5 of a mile and 5c each additional 1/5.
Factual error: In the scene where Kong is running around the streets of New York City and being chased by two military vehicles, the lead open-topped sedan has an air-cooled 30 caliber machine gun mounted on a bar behind the driver, with the gunner in the back seat. The time setting of the movie is 1933-1934. Air-cooled ground mounted machine guns were not in general use until 1942, and were not available for aircraft until 1936. The Army used water-cooled machine guns until the start of World War II.
Continuity mistake: When Kong falls through a portion of the circular observation deck at the top of the skyscraper, he crushes the metal siding as well as one of the metal girders, making a large hole. When he puts Ann inside through the hole, the metal girders are now all undamaged. You can tell that this is the same hole because the camera moves around the entire building in the next shot and there are no other large holes.
Audio problem: After the scene where the men had to fight off the giant bugs, there is a wide shot where the men can be seen being helped out of the pit by the captain and a few extra crewmembers. The captain is shouting something, however, if you look closely, his mouth is not in sync with what he is saying.
Continuity mistake: Shortly after Kong realizes he's been shot and gets blood on his fingers by touching the bloody hole in his chest, in subsequent scenes his chest has no puckered indentation and is unblemished but for minimal straight-line scars.
Continuity mistake: When Carl falls at the beginning of the brontosaurs chase we see that he is only just in front of the brontosaurs so when he fell he should have been crushed straight away but he isn't. Jack then runs to help him and we see that the brontosaurs only need another step or 2 (which would take 1 or 2 seconds) before they reach Carl and Jack. Carl gets up and argues with Jack for 5 seconds or so by which time they should have been crushed. They then run and there is a far off shot showing the brontosaurs back where they were at the start again.
Continuity mistake: Ann is soaking wet when she is being led through the natives. When she gets to the top of the wall she's dry.
Answer: I assume you mean, could the building take his weight, not whether an ape would really have the ability to climb a building (if that's what you mean, then it's definitely yes...apes are great climbers). Assuming Kong is proportionally as heavy as normal-sized gorillas, which tend to be in the area of 160kg (~350lbs), then he weighs over 80,000kg (89 tons, give or take). The average human weighs about 62kg, so that's about 1,300 humans, and the capacity of the ESB is over 13,000. So, assuming the building is mostly, or even half, empty while a giant gorilla scales it, the building could handle his weight.
Keep in mind, though, that the weight allowance for the building assumes people on the floors of the building, not climbing on the outside. The outer structure of a building isn't designed for massive creatures climbing on it. While the building as a whole would likely survive, there would be significant damage as Kong would be breaking windows and pulling stone off it as he made his way up.