When Prax's suit air hose becomes unplugged he immediately has no air. But Amos unplugs his and has plenty of air to reattach Prax and carry on. He should be out of air as well.
Each time Draper reduces the Razorback's acceleration in order to prevent Deputy Undersecretary Avasarala from passing out it shows the Razorback slowing down. However in space the ship would not slow down when the engine output is reduced, it would keep moving at the same speed unless another force acted upon it.
Gravity in all scenes on Io is too high. In reality, Io's surface gravity of 0.183 G is only slightly higher than that of Earth's moon Luna of 0.165 G.
In the distress signal sent out by Bobby Draper, she leaves her and Chrisjen Avasarala's names out of the transmission as both could be sent to prison. However, the signal received by Alex and Naomi includes the line "MMC Operator: DRAPER, A". This also means Alex and Naomi would know who they are rescuing when informing James Holden of the signal.