After Micah lands with his balloon, a shot from above shows him exiting the pod and the door bounces back shut. In the next shot, the pod door is still open.
At the beginning of the movie, Sam takes a sample looking to see if bacteria can use ammonia (NH3) as an oxygen (O2) source. Ammonia has no oxygen available to be used.
Ammonia is lighter than air, you would not find larger concentrations in subways or try to escape it by living on top of a mountain.
When Sam and Mica find the newly hatched queen bee inside the ruined beehive, it has a large white dot on its thorax. In reality this mark is painted on by beekeepers after she is mated, and should have never been there.
The movie states that the air is toxic mostly due to large concentration of ammonia. A detection method for such toxicity is naked flame of purple color.
Ammonia does not make the flame purple, but yellow (in very large concentrations). There would have not been an effect as described in the film. Ammonia is also very easy to recognize due to its unique smell even at concentrations below toxicity, so no such detection method would have been necessary.
Similar to the goof involving a lack of oxygen to run the ATV's internal combustion engine, the gasoline in it would go bad after a few months. Based upon the level of vegetation shown overtaking various objects, the gas in the ATV as well as any other that was available would have gone bad long ago. This along with car batteries that retain their charge indefinitely (a character gets in a car that has been sitting for months or years and it starts right up) is a common error in post-apocalyptic programs.
While looking at all of the dead bees, a bug flies by the camera. If all of the insects were dead, this bug would have also died.
When walking around the toxic zone with insufficient oxygen for the entire trip, Sam always leaves her lifesaving & lightweight backup tank with her ATV rather than carry it in her backpack.
One of the last people on Earth and living and working alone, with no thought to fashion or makeup, Sam still keeps her legs and armpits clean shaven.
Micah uses a rope to haul up the wind turbine, but there is no tension on the rope.
After landing his balloon Micah picks up a hammer, presumably to secure the balloon with guy ropes, yet he wasn't carrying a hammer when he stumbled out. He is then seen walking off without using the hammer or securing the balloon, although much later the balloon is now secured with guy ropes.
In such a low oxygen rich environment, the ATV's that were ridden would not work due to the engine needing oxygen to complete the combustion process.
Contrary to the claims that ATVs shouldn't work due to no oxygen, the movie implies that Sam needs the oxygen tanks to breath because excessive ammonia in the atmosphere has made it toxic. It's not the lack of oxygen but the presence of ammonia that is deadly.
When Sam is reading the letter from Elon, it states the exoplanet Proxima B(which is the correct name). However, when hearing Elon read this, he says instead Centauri B.
Sam asks for Micah's help to fix the wind turbine, then she just stands and watches as he struggles to erect it.