A family of Microsoft word processing software products for creating web, email, and print documents.
The AI's answer is more or less correct, if a bit jumbled (some of the numbered steps are merely introductions to the detailed steps that follow them). The key points are:
- Each section contains three headers and three footers (First Page, Odd Page, and Even Page, where the Odd Page ones are the default if the others aren't manually turned on). Each of the three headers is independent of the other headers with respect to contents and the "Link to Previous" setting. The same is true for the three footers.
- You said that when you turned on First Page headers, "The headers disappear on the chapter openings, but the page numbers remain." That implies that the page numbers were present (just not displayed) in the First Page header before you turned on Different First Page; you had been looking at the Odd Page header before that. You should be able to delete that page number to empty out the First Page header, without affecting the page numbers in the rest of the pages in the section (those in the Odd Page and, if you use the Even Page header, there also).
- If your blank pages are ones at the ends of sections, they may occur because you used Odd Page section breaks, so Word is adding those blanks to force the next section to start on an odd-numbered page. Even then, nothing should appear in the headers of those pages. There is a different way to make those pages blank, by creating a field that will suppress those page numbers, as described in https://wordmvp.com/FAQs/TblsFldsFms/InsEvnPgEndChap.htm.
If you still need more help, create a document that illustrates the problem (it doesn't have to be actual text from your book, as long as the layout is the same), store it on a service such as OneDrive or DropBox, and post a sharing link here so we can download it.