Answer: The "Archive" button removes an item from the content tree, but does not delete it from the database - if the item is published, it will remain visible on the live site until the parent folder is republished. Archived items can be restored at a later date.

NOTE: Agencies can archive pages, however it should only be used when an item is intended to be archived permanently, as restoring anything requires assistance from CI.
Learn more about the Content Editor Ribbon.

Answer: The "Hidden Items" checkbox shows some fields in the content editor which are hidden by default. Based on permissions level, it may or may not actually cause more fields to show up. It does not change the visibility of any items themselves either internally or externally.

Learn more about the Content Editor Ribbon.