Now that you bring this up, I'm baffled as well. I haven't customized a stair so much until I tried just now. I played with all the settings in the Run Type for each of the stair types: assembled, cast-in-place and precast. None of these settings made the nosing profile visible in plan view, even when I used an obnoxious 3" x 6" nosing profile extending above the treads. However, as you may have found, the Run Type is able to be selected in plan view, it merely doesn't have permanent lines in plan view, so to speak. See the images below (the instances are identical, I simply copied one stair over).
Plan view with Run Type selected
3D view
As for your first question, yes I am having this issue as well. As for your second, I'm thinking the system family for stairs has some symbolic lines and/or masking regions applied to its plan view which we, the end users, cannot change. Unfortunately I have no solution besides drawing filled regions and constraining them to your stairs (which I readily admit is the worst way to do this if other solutions are present).
I will be eager to hear if you find a viable solution or workaround.