Towards the beginning of the episode, Captain Pike asks the computer to show him Starfleet's rendering of the seven signals, then to 'rotate 90 degrees counterclockwise'. He doesn't specifies around which axis, yet the computer rotates it around the vertical axis, even though it could have been another one. Also the rotation starts exactly when he begins to pronounce 'counterclockwise', so the computer couldn't know at this time, in which direction to do it.
In the church basement, Pike mispronounces Owosekun's name as "Owosekin."
When Saru announces that the distress call has been on a loop for over 200 years, Burnham states that was "before warp was even invented." While it's true that it would predate humans' discovery of faster-than-light travel, there are many civilizations in the Star Trek universe that possessed warp technology long before humans. The Vulcans, for example, were warp-capable far earlier than humans, and Burnham grew up on Vulcan.
The landing party (consisting of Captain Pike, Commander Burnham, and Lieutenant Owosekun) were armed with #2 phasers (the ones with a pistol grip). They should have been armed with #1 phasers since they were in disguises, as #1 phasers are much easier to conceal.