Does anybody know whether the Footnote and the Index page styles are applied automatically under some circumstances, the same way as the Endnote page style is applied to the newly created page when we insert an endnote?
Your question is not crystal-clear. Please edit it to improve it (don’t start a conversation, this would rather create confusion).
A footnote is inserted at bottom of current page; it is laid out in a portion of this current page under the current page style; no page style change.
An Index (TOC, Alphabetical Index or other) can be inserted anywhere and be part of a page (e.g. you can insert a “chapter TOC” at start of every chapter; such a TOC is usually short and don’t deserve a page of its own); TOC/Index insertion does not cause page style change.
Thanks. But what is the reason to have the “Footnote” page style?
Well, Writer allows weird things: if you configure it adequately, “footnotes” become “endnotes” but are not the same as “official” endnotes.
Footnotes & Endnotes, the configuration dialog for notes. In
Footnotes tab, you can request your footnotes to be laid out at end of document.
The footnotes will then be collected and spat out in their own Footnote page(s), before the endnotes in their Endnote page. Both lists are not merged; they remain different despite having the same characteristics now.