How do I change the name and unit of the X-axis in “plot across design points”

If you don't want it plotted against the design point, then don't use "plot across design points"!

If you use "Plot All" instead, it may well plot with the x-axis being your swept variable. It might not though if you are sweeping more than one thing (e.g. corners or another variable) but you can always use the right mouse button menu over the x-axis to change the swept axis, or alternatively send to the calculator and use the swapSweep function. If you want such a swapSweep output to appear in your ADE outputs, you can generate the expression and add it as an output with the EvalType column set to "sweep" (in Explorer this column doesn't show by default; you have to go to the column headers in the Outputs Setup and right mouse button to enable it).

Andrew