Factual error: The whole conspiracy is doomed from the start! They show a Lunar Module on the set representing Mars; the astronauts were supposed to use this to 'land' on the Martian surface. This is insane. The Lunar Excursion Module cannot operate in an atmosphere; it would burn up on entry. It has no heat shield - the engine nozzle and landing legs would have to poke through it if it had. Exterior protrusions like antennae and steering engines would burn off during descent through the Martian atmosphere. While Mars' atmosphere is thin, it is still more than sufficient to burn up a Lander designed this way. All three recent Mars Landers, Spirit, Opportunity, and Phoenix, had heat shields and parachutes to facilitate safe re-entry. This is not a radical redesign of the LEM; it simply cannot be the shape it is if it is to work in an atmosphere. Any schoolkid would know that, and the world's media would not be fooled for a second.
Factual error: If one person (at NASA) was able to calculate the distance between the returning Mars capsule and Earth based on their radio signals, then everyone with basic radio receiving equipment would be able to. The Russians routinely track American spacecraft and publish details of their trajectory in Pravda, and amateur astronomers throughout the world do so, too. This all comes from triangulation of their radio signals. There is no way that NASA could hide this simple fact from their worldwide audience.