Thought you might be able to help me with this overprint problem. I prepared some materials for offset printing with indesign. I used 100% Black for the color of the fonts. When it was printed out, the black color on the white background looks good. However, some of the black on the colored backgrounds are not black enough. It seems that the black was printed over the colored backgrounds and caused some of the blacks to fade. When I ask the printing guys, they said it is called overprint and it happens if you use 100% black.
Your printer should be responsible for trapping and overprints(that is what they get paid for). Sounds like your printer needs help in figuring out how to turn off overprint.
To take black overprint off of the entire document
Edit >> Preferences >> Appearance of Black
Uncheck "Overprint Black Swatch at 100%"
To check overprint is not assigned to an individual object
Window >> Attributes >> make sure overprint fill is unchecked
To test your seperations
Windows >>> Output >> Seperations preview
Check view seperation, then hide all the eyeballs except the color you want to preview
One thing confused me about your post, as where the black overprinted, you siad it faded. It should have been glossier/darker with the extra colro underneath. Maybe by faded black you meant that it had a color tint to it.
A rich black is composed of 100k 20c 20m 20y or up to 100k 60c 60m 60y. Basically the paper soaks up the ink and adding some extra CMY in somehwat even amounts helps make a black darker.
In regards to rich black, you might want to ask the printer (a loaded proposition, I know) what formula works well on the intended press and stock. There is no set or "magic" formula and the *should* know what works best. If they throw a lot of printing terminology at you, just ask them to clarify...it's in their best interest, after all. In most cases, I prefer to have black overprint...because black will cover most colors/inks and you avoid registration problems. I would only not have it overprint if there is a clear cut reason for it. Hope that helps!