It should automatically keep the calibration (as well as other settings). You can alter some values in the custom settings menu and see if those are being saved. If not, perhaps you can investigate so that there is no permission issue that keeps it from saving (I'm not familiar how these things work in Linux). (In linux I believe the Playerprefs are stored under "/home/your user/.config/unity3d").
As you have sort of figured out you select the mode during calibration (by using the stick that you want for each axis, for example, it says "throttle up" on the top of the calibration window and you use the stick you want to assign as throttle. It is described in the manual. The quick mode selection is for touch screen input only). I remember another Linux user reporting a while back that he also had to recalibrate each time. Might have something to do with the joystick drivers if that is the case for you too.
