You needn't touch the original skirts. The Cup skirt fits over the original leaving the part of the originals in the door aperture visible.
Exactly how the Cup skirts attach is unknown to me but I suspect that the screws underneath ( longer ones needed? ) are re-used to attach both skirts to each side of the car. Adhesive for the upper part?
Cup skirts go over the top of the original sill covers.
You are correct in your assumptions Jon.
You will need to first remove all the screws from the sill 'turn-under' that pass through the original plastic sill covers.
The screw heads face towards the ground, on the other side and removable plastic threaded piece creates the clamping effect.
Due to their location Expect the following: corroded disintegrating screw heads, AND the plastic threaded part to be brittle causing it to break away from its snug mounting hole causing both to just spin, swearing will ensue.
If you get the screws unscrewed the plastic inserts can be removed from the back of the sill too, highly likely you will need to replace quite a few if not all of them.
Clean the plastic sill covers up with some alcohol or solvent based cleaner.
I would then apply black 'body kit' sealant, mostly on the the 'fillet' edges (ie where it wraps around the original skirt on the leading and trailing edge of the sill to wheel arch, along the tops either side of the door.
The section with the exposed Alfa Romeo Logo on the sill cover I would put a trainline style bit of sealant around the cutout in the Zender skirt to get the best seal.
This will stop grit dirt and damp getting between the two plastic covers, this is important has hard debris between the two could potentially cause damage to either item.
Once applied and sealant still soft, work your way along the 'turn-under' and start fitting and tightening the screws.