The correct sequence for the Device's Memory Cores are randomized every run-through, so determine through trial-and-error.
Once the Device is in Astrometrics, do a "encrypted" intrusion.
The Breathing Apparatus are not available until halfway through the "Prison Break." Bring plenty of EV Suits before you even start the mission, or suffer Holds every so often. However, don't equip the EV suits if you want to get the Breathing Apparatus. Equipping the EV suit removes the option to get the Rebreathers.