The Funeral - S1-E3
Deliberate mistake: Though it serves the drama and black comedy of the scene, it's highly unlikely a modern, let alone future broadcasting network would neglect the importance of delay (usually around 30 seconds) between recording and "live" broadcast. Specifically, to prevent unexpected gruesome incidents like the exploding head seen in this episode from making it to air.
Factual error: The communication delay for earth-moon communications is around 2.6 seconds. Yet Emma's daughter simply dials a cell phone and is connected to her Mom in space past the moon in less than that. The communication has no time lag. (00:18:45)
Character mistake: Dr. Agnes Jurati, one of the Federation leading scientists, is bored during the hyperspace travel, and so she chats a bit with the captain. While she gives her quirky speech, she casually mentions that "there are over 3 billion stars in our galaxy." She's not wrong, technically, but the number of stars in our galaxy is estimated between 100 and 400 billion. She is way off. (00:08:40)
Suggested correction: Two things. You state that she's not wrong, which she isn't. The fact that she chose an odd turn of phrase doesn't make it a mistake. Plus, you reference the number of 'galaxies' in our galaxy but I am guessing this is just a typo.
Revealing mistake: As Ed slowly backs away from the camera (from the foreground to the background), the towering Scrapper robot moves slowly forward (from the background to the foreground) to take Ed's place facing the camera. The lumbering robot is plainly casting a long shadow across this night-time shot towards the right side of the screen. But, as Ed and the giant robot pass one another, the CGI robot's shadow never falls across Ed at all. Throughout the shot, Ed remains uniformly illuminated, revealing that the digital-effects artist neglected to add the shadow passing over Ed. (00:38:55 - 00:39:19)
Episode #1.7 - S1-E7
Other mistake: Surveillance cameras are shown on monitors when the mass panic kicks in even though any chip-dependent electronics shouldn't work in the game world (previously stated by Chotu). (00:08:20)
Factual error: In Massachusetts, Atticus, Leti and George are attacked by the local sheriff and his deputies, who are patrolling the county roads. In Massachusetts, sheriffs run the county jails, but they do not (and did not) handle routine policing, which is the responsibility only of municipal police departments.
Factual error: Paul tells Shea that he has a condition called Sporadic Fatal Insomnia. Later, Shea finds a fork in a bag labelled Abby. Paul explains that Abby is his daughter, and he wants to test her DNA, because she has a 50-50 chance of contracting the disease. This is false. Sporadic Fatal Insomnia is not hereditary. Fatal Familial Insomnia is genetic and is the inherited form.