Book structure translates into pages styles.
From your description, you need at least 3 pages styles: cover, TOC and running content. Page numbering is an attribute of page style. Cover has no numbering. You can number your TOC with lowercase roman (common convention) starting from 1. The running content will be numbered numeric starting from 1.
Restarting numbering is triggered from the page break causing the switch to the required page style.
Header and footer are also attributes of page style. Cover has none. TOC has specific ones while running content usually display a reminder for book title and/or chapter heading (this can be automatically inserted with fields).
Remember that pages do not exist as such in LO Writer. They are the result of text flow overflowing the available space defined by the current page style. Overflow may cause an automatic switch to another style if correctly configured.
Concerning your inability to change page style on first page, we need more information. Edit your question or attach your document.
EDIT 2018-05-01 Analysis of attached sample file
No problem to change page style on first page, however your document is badly structured (use of direct formatting for changing style in page breaks?).
First remember that built-in page styles have identical properties as shipped. To exhibit style change I set top margin to 10 cm in First Page only.
With this setting, your document alternates between First Page and Default Style. Default Style is the follow-on style of First Page (default setting), but Default Style → First Page must have been set by you (Organizer tab in page style definition). Restore Default Style to cancel alternation.
Unless you oversimplified the sample file, the main problem in your document is the lack of usage of adequate paragraph styles. For example, I don’t understand how you obtained your TOC because no paragraph is styled Heading x: all paragraphs are Default Style. Some of them have a page break manually added.
You should define ad-hoc paragraph styles corresponding to semantic usages. This simplifies document maintenance.
In addition, the styles should define their “private” spacing above and below their text. Using empty paragraphs to create spacing is wrong: if you change margins you can’t guarantee you will keep in sync with pages. In the same way, if you want multi-line headings and enter them in the TOC, use newlines (Shift+Enter) – newlines don’t break paragraphs.
Obviously, your Part headings should be styled Heading 1 and your chapters Heading 2. Both styles should include a forced page break to some “common” page style (maybe Default Style if you dedicate it to running text).
Your table of contents should also be laid out in its own page style which could be triggered by your custom TOA Heading paragraph style. As a side remark, the heading for the TOC can be generated from the dialog in Insert>Indexes & Tables>Indexes & Tables instead of being typed manually before the TOC. Using the feature guarantees that the heading always follows the TOC, should you decide to move it somewhere else.
With such an important planned document, styles are your friends. Basic styles are those for paragraphs. They operate within pages. If you need to locally override paragraph style (part of a paragraph, otherwise it is a new style), use character styles. I know by experience that designing a satisfactory set of styles is difficult but it’s really worth the pain because you simply focus your attention on content afterwards, not on layout. Moreover, with such styles, you end up with really less typing.
