As Adira climbs down a ladder to talk to Stamets, his head is tilted to the right, but in the next scene, Stamet's head is tilted to the left.
Human hair grows at an average rate of about a half an inch per month, or about 6 inches per year. Burnham's hair is shown growing exponentially faster than that, reaching a length of several feet in just 12 months.
Discovery jumps into Earth's solar system and arrives next to Jupiter. They then arrive at Earth mere seconds later. Since they had planned to travel at sub-light speeds to Earth once arriving in the solar system, they would have been traveling using their impulse engines. Full impulse speed is 1/4 the speed of light. At that speed, it would have taken approximately 172 minutes to reach Earth.
When putting on uniforms to blend in, Booker says "Aye Aye, Commander" and Burnham says "One aye, we're not pirates." In the Navy, one "aye" is for "understood", the second "aye" is for "will comply".
Michael Burnham's opening log entry is cut off as the communicator beeps, as though she were recording it at that moment. But she is seen sitting and laughing with Book when the communicator beeps, so the log entry must have been recorded at a different time and suddenly cut off for no reason.
Burnham insists that Book dress in a Starfleet uniform when their ship is boarded, as anyone not in uniform will be suspect. This ignores the fact that, on active duty starships, the ship is manned on rotating duty shifts. At least 2/3 of the crew is off-duty at any given moment and crewmen are often seen out of uniform during these times.
Michael suggests that Discovery jump to a point outside Earth's scanning range and pose as a starship that was stranded by the Burn which has been attempting to make it back to Earth at sub-light speed. A ship stranded by the Burn would be just over 100 years old or so. As soon as anyone from Earth scanned them, it would be immediately obvious that their ship is of a much earlier design and registry, nearly 1000 years old. Scans would also reveal that they have an active warp core, something which would also contradict their story.
Michael informs Book that what he's been drinking in synthehol. Synthehol was an alcohol substitute introduced some time in the 24th century, as established in Relics (1992). Discovery, being from the mid-23rd century, would not yet have been familiar with the substance.
Book complains that he's been trying and failing to "get a buzz" from synthehol. Synthehol is an alcohol substitute that affects the body in the same way as alcohol but whose negative effects can be easily dismissed (how this works has never been explained, but that is the in-universe explanation). It should be easily possible for Book to "get a buzz" should he so desire.
Burnham and Book depart Discovery in Book's ship. After the bridge crew has a lengthy conversation about their plans, Book's ship is shown to have only made it a few meters out of the shuttle bay.
Stamets refers to Adira as a "tween menace." A tween is an adolescent that has not yet reached their teenage years, typically between the ages of 10 and 13. Adira is stated to be 16 years old and therefore by no definition would they be a tween.