Back again, just had a read thro all your comments and a play in SU. First up, a big THANK YOU to everyone - I finally get it (well, some of it some of the time, anyway).
Predictably, it turned out to be user error - thanks Dave for having the patience to step through the procedure, and jeff H for the video which rammed the point home. For the sake of other newbies like me, the error I was making was not intersecting the faces. So in Daves skp file (see upthread), I extruded the cutouts down past the model and assumed that this act alone created the intersection - it doesn't. You have to select everything, then right click - intersect faces (this puts the 2 elements, cutout and main model, into the same context, yes?) and NOW its easy (see Daves pics & Jeffs video). You have edges where you need them to delete the outside stuff you don't want. It takes seconds to clean it up. Jeffs method is kinda the inverse of this, and it also works really well. I've spent hours trying to do this, when in reality it should only take seconds - if you know how.
So now I know 4 seperate ways of doing this, 2 with inbuilt tools and 2 with plugins. I really appreciate everyone taking the time to offer help and advice.
I have a related question, prompted by some stuff upthread - if the mods think it better to make it a seperate thread, no problem.
Gilles said "When you proceed to deleting unwanted geometries, make the part soft and smooth", and Box mentioned welding the segments of the curves together. It got me thinking, because the ultimate aim is to print a real clip on a 3D printer (I'm new to that, too.....).
Now I know that SU draws circles as a series of straight lines, and I wanted the final 3D printed object to be circular not blocky, so I increased the segment count on the half circles to 100. Is this sensible, or just way too much? Gilles mentioned smoothing - does that just improve the visuals aspect of the image in SU, or does it have a real impact on a 3D printed object?
And Box mentioned welding the lines/segments of the curves together (presumably because the model was exploded at some stage in its history). I googled about this, and got rather confused. I understand that welding the lines together creates a polyline (polygonal curve), but I'm not clear how that actually differs from a regular line, and whether the arc/circle tools in SU create a series of regular lines or a polyline. And in the context of 3D printing, is it better that the curves be default SU arc/circles, or polylines?
Phew, another whole bunch of questions. I would like to finish with another, very sincere thanks for all the help you guys have given.