Word stores who created a document and who saved it last, often with the company name. Reviewers' names are stored separately, in comments and a reviewer list. See each of them and choose what to remove.
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Author, last modified by, company and the reviewer list are removed with one choice each. Names on comments and tracked changes go away when you remove the comments or tracked changes. To keep comments but hide who wrote them, add the reviewer's name under "Add your own private terms" and choose Label or Pseudonym — the name is replaced wherever it appears, including on the comments.
The author property is only one place. Reviewer names are stored with every comment and tracked change and in a separate reviewer list, which remain unless you remove them too.
If you edit and save the cleaned copy in Word, Word writes your name as Last modified by again. Clean the final version just before sending.
Yes, when it is set in the document's properties. It is shown as a separate item you can keep or remove.
The new copy is reopened and scanned again; each removed item is only marked as verified if the second scan doesn't find it.