The Apollo missions did not use space shuttles. The vessel consisted of a Command Module, which is what the astronauts travel to the moon and back in, also is used for re-entry, and a Lunar Module which is a two man craft used to land on the surface of the moon. They were launched into orbit using a Saturn V three-stage rocket.
The lunar module set contains numerous errors, including the use of seats (which were eliminated from the real ship to save weight) and the size of the interior (the original was barely able to hold the three astronauts when they used it as a life boat).
In order to save weight the LEM did not have seats and the astronauts stood for launch and landing. This was possible due to the moon's gravity being 1/6th of earth's. Also, the LEM would be about half the size depicted on the show.
Buzz Aldrin has stated that the flag they left on the moon on Apollo 11 mission was knocked over when they blasted away from the surface with Lunar Module. In the episode its standing upright with nothing else around it when Ray Palmer goes to retrieve it.
When Ray retrieves the flag from the Apollo 11 landing site, the bottom stage of the Eagle lunar lander should have been sitting close by, but it is nowhere in sight. Also, the flag should have been lying on the ground, as it was knocked over by the engine blast when the astronauts left the moon.
It is revealed that Eobard Thawne was impersonating one of the Apollo astronauts. However, Thawne returns to Earth aboard the Waverider, meaning the Apollo spacecraft would have returned with only two astronauts aboard. This would be impossible to cover up, as all Apollo spacecraft recoveries were highly televised events.
The Atom suit often withstands gunshots, energy blasts, etc. However, here it can be easily damaged with a small knife.
The wave rider used the tractor beam to slow and stop the command module, then it just sat and waited for the meteorites to hit it. After it is damaged, it then moves away. Since the module had been stopped, there was ample time for the wave rider to move out of the way, but it did not.
It is stated repeatedly that Reverse Flash can't use his speed in zero gravity. Yet he uses his super speed shape shifting ability in the Apollo shuttle.
From 32:24 -instead of worrying about the proper reentry angle just ask one of the NASA scientists in the room.