“This book is part of a series”
The first part of this section here should be self-explanatory. If the book is part of a series, check the box. If not, leave the box unchecked. Why does this part matter? As Amazon says:
Books in a series with the same series title will automatically link and create a series landing page on Amazon.com, in order of volume numbers. This improves the discoverability of all the books in your series by showing customers all of the books that make up the series in a single page.
So that’s something you definitely want if your book is indeed one in a series.
Here is what Amazon has to say (if you hover your cursor over the “What’s this?” link next to the edition number field) regarding the edition number:
A version is a particular edition of a book. Identifying the version number helps readers know whether the book is the original edition, or if it contains updated content. If this is the first time you have published this book, enter the numeral 1. If the book was previously published and the version you are publishing contains significant changes, enter the numeral 2 (and so on).
It is up to you as the author to determine whether the changes you are making are “significant.” I would not treat typo corrections and other minor fixes or updates as new editions.
You don’t have to specify the name of the publisher. It’s optional. If there is a real publishing company behind the book, the name of that company should obviously be entered here. If you are self-publishing, you can just make up a publishing company or enter in your own name. I’ve also seen some authors list websites here (e.g., “JohnDoeAuthor.com”), though Amazon discourages the practice (“Only the names of the author or publisher of the book should be entered in the publisher field.”).