Lorelai is not wearing the same bra when her body is discovered as when she jumped into the water.
Lorelai's dress has cutouts that show her skin, except in the opening scene when she dives into the pool, when the cutouts are backed by white mesh fabric.
James re-caps his hooch bottle before throwing it into the pool, but a closeup of the bottle shows it without the cap.
After the "radiation alarm", the green shirt of Dr. Bryce is closed at the top, then open while she speaks with Nora, then closed after Nora is gone.
The bullet was identified as a .22 short. The firearm recovered was a Ruger MKII/III semi-automatic. The MKII/III is chambered of the .22 LR, or "Long Rifle" round, not the .22 short. But, the MKII/III, like other guns chambered for .22 LR will fire the .22 Short round. The round is not powerful enough to cycle the action and does not fit the magazine to feed reliably. The .22 Short will work reliably in a .22 LR revolver.
"...a .22 short. Not a common caliber, even on Earth." Untrue. The .22 Long Rifle is an evolution of a series of .22 rimfires. While no .22 Short firearms are made today, they are still around, and were fairly common in the 1960s. In addition, from the bullet alone it would be impossible to determine what type of .22 cartridge it was, other than some rimfire cartridge of that caliber.
During the radiation storm, Captain Denninger plays "Stormy Weather" on a phonograph by placing the needle on Track 1. However, that song was on Track 6 of that particular Lena Horne LP.
The whole plot-line about the uprising because of a point of no return doesn't make much sense. At this point in their journey they have accelerated 50 years. At the halfway to their destination they turn the ship around and decelerate by simply accelerating in the other direction. Even if they turned around a few months early on journey of this scale they would need 50 years to reach a standstill and another 100 years to get back to earth - much longer than the 50 years to their destination.
TV and radio signals from earth have reached 74 - 110 light years distance, depending on transmission dates and media. This means the 'Ascension' could pick up these signals. The argument could be made that they didn't have the technology. A super advanced trillion dollar star ship, which can travel between stars, but they can't receive free-to-air TV or even radio.
At 1:40, as medics are recovering Lorelei Wright's body, Amanda Thomson can be seen clearly moving her arm in the overhead shot.
(at around 21 mins) Nixon, the electrician, uses a multimeter with an LED digital display. LEDs had barely been invented at launch time in 1963.
After the death of Lorelai, Dr. Bryce is seen examining the body wearing latex (or similar) medical gloves. In fact, medical gloves were only invented in 1964, a year after the supposed launch of the USS Ascension. Even if gloves were available, it did not become common practice for medical professionals to wear gloves during routine examinations until the 1990s, triggered by a fear of pathogens being passed on through minor contact with blood or other bodily fluids. As such, it is highly unlikely that a doctor from a culture rooted in the early-60s would use gloves. Indeed, the premise of the show is that no murder had occurred in the first 51 years of the mission, so a doctor would have no reason to think to use gloves to examine the dead victim of crime.