Corrected entry: After Gemini 8 reaches Earth orbit, Neil Armstrong tells Houston "We have SECO." SECO is Saturn Engine Cut-Off, which would be the appropriate term on an Apollo flight, as they were all launched on either Saturn V or Saturn I-B boosters. The Gemini missions, on the other hand, used a modified Titan ICBM as their launch vehicle, so speaking of a "Saturn Engine Cut-Off" in the context of a Gemini flight makes no sense.
Correction: That's because SECO does not stand for Saturn Engine Cut-Off. In any launch (you can watch them all the time now, since most US-based launches are live-streamed) they always have the callouts MECO and SECO, standing for Main Engine Cut-Off (first stage) and Second Engine Cut-Off (second stage). NASA's own list of acronyms confirms: https://www.nasa.gov/pdf/140648main_ESAS_17a.pdf.
Corrected entry: When Neil Armstrong is talking to his children before he left home for the Apollo 11 launch, if you look at his wrist watch (which can be seen various times) the second hand doesn't move at all right through the scene.
Correction: The watch he is wearing is an Omega Speedmaster. What you think is the "second hand" is actually the hand for the chronograph and only moves when the chronograph is actuated. The running seconds hand is in the 9 o'clock subdial, which is too small to see.
Corrected entry: When Neil Armstrong is walking on the moon, he stops to drop the bracelet of his daughter. This small, plastic bracelet simply falls to the ground. With no gravity, it should have floated away.
Correction: There is gravity on the moon, albeit very much less than Earth. Moon's gravity is about one sixth of that of Earth so the bracelet will actually fall to the ground at the respective velocity.
Correction: The moon has gravity - that's how he's walking around! There's just lower gravity than on earth. I mean...there's videos on youtube of astronauts on the moon dropping stuff, falling over, etc.
Corrected entry: When Neil Armstrong has tears running down his face while standing on the Moon, the tears should fall much slower as the Moon's gravity is approximately one sixth that of Earth.
Correction: The first tears would travel slower due to the reduced gravity, however after there is moisture on his face any further tears would travel down much faster due to cohesion between the water molecules, in a similar way to how water on a window will try to clump together.
Although the tears do get faster due to cohesion, the maximum acceleration should not be greater than one sixth than that of Earth. However, it is seen to be travelling as it would on Earth.
Since the air inside his lunar suit is very dry (I doubt they add humidity), I would think the tears would not get very far before evaporating.
Correction: "We have SECO" was part of the actual Gemini 8 transcript. It can be found in the mission transcript, 3/16/66, 8:58 a.m. Tape 20.