Erik suffers from diabetes and says he urgently needs his insulin (which is used to lower blood glucose levels). The insulin is thrown away by one of the good guys a couple of minutes later and then Erik dies in a minute or two because his blood glucose has fallen too low (if that was the case why was he urgently needing insulin which would lower his blood glucose level?)
To be able to go outside of the aircraft and remove the engine, they would have to descend from cruising altitude down to at least 10,000 ft. Climbing back to cruising altitude would likely burn off way more fuel than they would have "saved" by removing the drag and weight of one engine. (So would staying at the low altitude).
A pilot would not contact "ground control" while airborne. The ground controllers are responsible for controlling the ground environment at an airport. At cruising altitude they would contact the en-route, center or area control.
The air in a cabin is exchanged frequently through the flight but if they want to clear the air they would open the outflow valve, not the exit door. They could have also donned oxygen and breathed it to get fresh oxygen.
First they claim they have to descend to open the door, which is correct, although around 10,000 ft should be enough they seem to descend to below 1,000ft. Later when they are to chop of the engine, they have no problem doing this at what appears to be cruising altitude, which would not be survivable.