Tony is performing the first astronaut space walk, but he is doing it from a 3-man, Apollo capsule. The first American spacewalks were done from the 2-man, Gemini capsules.
In Season Two, Tony does another spacewalk from a Gemini capsule, with Roger as his co-pilot. There were no Gemini missions that came after the first Apollo mission, and none of the Apollo missions involved EVA spacewalks.
In Season Two, Tony does another spacewalk from a Gemini capsule, with Roger as his co-pilot. There were no Gemini missions that came after the first Apollo mission, and none of the Apollo missions involved EVA spacewalks.
When Tony is leaving the capsule for his space walk, Captain Healey's helmet is still open during depressurization and while Tony is outside the capsule with the hatch open.
When Jeannie, in smoke form, enters Tony's jacket's pocket at around 6:50, the device which is emitting the smoke can been inside the pocket. The device presumably emitted the smoke, and then scene was then shown in reverse so that the smoke would appear to go into the pocket rather than out of it. It was likely something that burned, but could have been a tube to emit the smoke or mist.
Nelson's spacewalk occurs while tethered to a three man spacecraft, which would not come into use until the Apollo program three years later. The first spacewalks were done during the Gemini program which only carried two astronauts (hence its name).
Near the end of the episode when the spacecraft is launched, it is clearly a Gemini capsule which would be a 2 man vehicle, once "in orbit", the capsule is an Apollo capsule, a 3 man vehicle.
Capt Nelson is performing a battery of physical tests when Dr Bellows directs him to the "pulmonary function tester". Dr Bellows should know better; the proper name for the device is "spirometer".