When Boimler is hit by the spear, it sticks straight out of his chest. When he rolls onto his side, it is still pointing straight up into the air, which would not be possible from the angle it hit.
When speaking with Captain Freeman, the tear on Boimler's uniform disappears. It is later present in all subsequent scenes until the end of the hunt.
K'ranch stabs into the side of Boimler's uniform, away from the front seam, yet the tear in the uniform is then depicted as crossing over the seam.
The way they're depicted here is not how 'space elevators' work. The upper terminus for travel is only the halfway point of the 'string'. There must be a counter-weight set as far out into space as the 'elevator' is high, otherwise the 'strings' would just wrap around the planet and crash to the surface. Also, the elevators must originate from the planet's equator to compensate for Coriolis disparity, and all the elevators shown are rooted in the mid-latitudes of the planet.
To finish out the Klingon game, Boimler enters a bar and demands bloodwine, to which the bartender rips off his arm and beats him to death with it. Martok declares this to be a dishonorable death, as he technically died by his own hand, implying that Klingons view suicide as a dishonorable death. It has been previously established, most notably in Sons of Mogh (1996), and even in I, Excretus (2021), that Klingons do have a system for honorable suicide.
The Cerritos takes all passengers from the broken orbital lift onboard until the lift can be repaired. There are numerous other lifts depicted around the planet. Given that some of the passengers had time-sensitive business to attend to on the planet, the Cerritos should have just transported them to one of the other lifts.