Without getting the actual file to inspect it is sometimes hard to say, but at first glance I would agree that setting all black objects you're talking about to overprint should be safe.
As for the white objects you mentioned: Make sure they are not set to overprint.
In general: Always check your files with View > Overprint Preview.
Thanks so much! (this is the best community!), so I'll set all the overlapping black (icons and text) to overprint; and I don't need to fuss with knockouts or anything?
Yes, do it.
If you've got further questions just talk to your print provider.
Remember that an overprinted black may sometimes show if it is printed over a partly-coloured background.
In some cases it may be better to use the colour of the background plus 100% K than overprinting the black.
I have deliberately desaturated the black to clarify my point, but this is well worth considering, especially when printing on soft papers.
So to fix this, the icon that is partially over a 40C 10K circle would entirely be 40C 100K ? and the icon over a 60C 20K circle would be 60C 100K? then all the icons will have partially different blacks - or maybe just make them all something like 20C 100K?
edit: i 'm working for the book packager, who has very little info from the printer, paper stock, etc.... but I could prob find out. how would I tell if it was soft or hard paper, and what would I want to hear to know if it was a problem or not?
You might get away with a sort of average mixture of cyan and black.
Don’t worry about paper stock to start with.