Artistic License - Physics: The astronauts are shown running across the lunar surface. For those unaware, running in low gravity is near impossible. It's best to hop along as shown in the genuine lunar landing footage. It's akin to running or walking in water.
Also, the notion that devices located on the surface of the Moon would be of any practical use for detecting Earth-bound ICBM launches. Not only would the Moon be on the wrong side of the Earth to watch for them much of the time, but the distance would impose a needless time-lag for the detectors to notice a launch and report back to Earth about it. This one is justified, though: the devices were never meant to do that, and the astronauts realize this when they take a moment to actually think about their mission.
Even if such devices could be made feasible, they'd need to be somewhere other than the poles, which never face Earth.
Several shots show the Earth a short distance above the lunar horizon. At the lunar south pole, it should overlap the horizon.
Covers Always Lie: The poster for the film depicts a three clawed footprint next to a human one. The aliens are spider-like, and don't leave conventional footprints.
Cold War: "The Russians" are mentioned. Turns out they made it to the moon...but they never came back.
Subverted. It's implied the USSR and the US were working together on the mission given how fast the Russians linked Benjamin Anderson to the Department of Defense.
In addition, the controls of the Soviet LK are almost identical to those of the American Lunar Module, suggesting a technology exchange which is reinforced by some of the viral marketing's top secret documents. In the Cold War setting of the film, it may be the product of espionage, but the actual Soviet LK was developed entirely independent of the American space program (and limited not by itself, but the failure to development a rocket to carry it at the time).
Downer Ending: All three astronauts die, and the government covers up their deaths so that no one will ever know the truth. Slightly subverted in that by watching the footage, you now know what happened.
Leave the Camera Running: Largely subverted, as apparently the movie was edited from 82 hours of Found Footage. While the movie may feel much longer, it has few moments not specifically dedicated to the plot, possibly to its detriment.
Make It Look Like an Accident: The three crew-members are all listed as having been killed in accidents at sea where their bodies could not be recovered.
Mission Control: The NASA kind, and a few seconds of an unspecified Soviet counterpart.
Sadistic Choice: At the end of the movie, Grey has to choose between saving the life of Anderson, his only surviving crew-member who might be infected with an alien parasite, or obeying the secretary of defense, who orders him not to, under threat of not being allowed to return to earth. Subverted in that both crewmen are killed.
Bonus points for including the Soviet LK lunar lander. The actual Soviet moonshot vehicle, the N1, was kept top secret until after the end of the Cold War, and even today not a lot of people know about it.
Also clever: since the N1 is a historic failure, the 1970s Americans call the Soviet lander "LK-Proton", implying it was launched on the Proton rocket (the same year, a Proton rocket carried Luna 22, the last successful Soviet Lunar orbiter, to its destination).
Space Is Cold: Actually justified for once. The crater investigated by the astronauts has never received direct sunlight.
Unwitting Pawn: It turns out that the government suspected there were aliens on the moon; they just needed to confirm it, which was the purpose of the whole mission. Needless to say, none of the astronauts were told about this.
Urban Legends: The movie is based on the premise that the stories of Lost Soviet cosmonauts are true—and that the U.S. government was not only aware of it, but actively involved in the failed mission.
Viral Marketing: Starting with an alleged moon program conspiracy blog entitled LunarTruth. It's even featured at the end of the film.