The Y input/axis becomes Z in lathe mode, so the Y0 sets it to zero. I use all three axis so the pretty logical that the the zero for each axis matches the keys (not necessarily the display). If you had a compound scale then you could sum the Z and Z1 axis or operate them independently. I use the 3rd axis for the tailstock which is more frequently used. Much easier then counting revolutions with the mechanical dial. I have checked the accuracy linearity of the magnetic scales and they track perfectly with the dial and gauge blocks over the full travel. Just double check that the scales track with your dials, the magnetic scales should be more accurate.
Diameter is just a button on the display R/D and it remembers the setting. Should change the color and show a little circle with a line through it. I find the white on black and the diameter in yellow to work the best for my eyes. Glad you have it mounted and running, should make turning much easier once you get use to it.