Other mistake: There is something drastically wrong with the design of the spherical 'Aries' moon shuttle. Some seats and many fixtures are 'upside down' relative to the up-down orientation of the shuttle itself, and we see loose food trays and equipment about the place as if this is routine. But - the shuttle is designed to land on the moon. What happens then? The moon has gravity, remember? There are going to be quite a few very disgruntled people dangling upside down like spiders, and there will be loose gear (and perhaps a stewardess or two) bouncing about all over the place. It is not a matter of stowing loose gear or lying flat on landing - some parts of the shuttle are upside down relative to others, which is why the stewardess has to do that famous 180 degree upside down walk. Whichever way you look at it the shuttle is going to encounter serious problems when it reaches a gravity well, which will occur whenever the engines are fired up, never mind landing on the moon.
Continuity mistake: When doctor Nemur, that other doctor, and Mrs. Kinnian are discussing possible complications with Charlie's surgery, there's one shot where Dr. Nemur takes the cigarette out of his mouth and in the next shot it's back in.