I have succeeded in getting a new section for my Chapter 2, with different headers for odd/even pages. However, the pages numbers are all the same as the first number in that section. This was not a problem in Chapter 1, although it was set up the same way: separate section for first page (so could have no header or page #), then another section for the rest of the chapter, with different odd/even headers. In that case the pages numbers remained consecutive, but not with this Ch 2.

That suggests you either typed the number or unlinked the field containing it. To fix that, select the errant number and choose Page Number>Current Position>Plain Number. As you're using a 'different odd and even' page layout, you will need to do this on both an odd page and an even page.

__________________
Cheers,
Paul Edstein
[Fmr MS MVP - Word]

This was helpful but now I have a new problem. Going along swimmingly, but suddenly, in a new chapter, the second page, the right-hand page, has an odd page header, when it should be even. Everything seems to be the same as in previous chapters: section change between the previous chapter and the first page of the new chapter (which has no header), then section change before rest of the chapter. Why would the odd/even pages suddenly be wrong?

Do not enter a section change before the rest of the chapter. Instead, use the option for different first page, which is a section setting.
Header/Footer Settings Recap