Does the new page count update my cover?

Yes. When you format your interior, the resulting page count is written back to your book, so the Cover Maker knows the real number and builds the spine width to match. This is the reliable way to avoid the most common cover rejection — a spine sized for the wrong page count.

The workflow that keeps everything in sync:

1. Format the interior first and note/accept the final page count.
2. Then export the cover, so its spine matches the finished interior.
3. If you later revise the interior and the page count moves, re-export the cover as well.

When the book is finished, lock its layout on the Export page. The formatter sets your book fresh each time you open it, so without a lock, simply opening a finished book can move the page count, and your spine with it. A locked book keeps the page count your cover was made for until you choose to allow changes. See What does Lock layout do?

See Why does the cover maker need my page count?