How to change the default search button value in awesome zymphonies free theme?
How to override the search text string

For how to change the text of the search box, you'll need to do a string override.  See the answer to this question on how to write your own custom module to do this.

Drupal also has a String Overrides module so that you don't have to write your own code, but it's not ready for Drupal 8 yet.  There is a test version here but it's not safe for production sites.

A third way to do this-- and perhaps the easiest approach if you will be changing lots of Drupal interface text-- is to add "English" as a translatable language on the language settings page (admin/config/regional/language) and then disable the original "English."  What this allows you to do is "translate" any text that appears in the Drupal UI.

How to show the search box on other pages

The search box is (most likely) a block that can be controlled on the block administration page: /admin/structure/block.

You can edit the block settings to choose which pages it appears on.  You can also place additional search blocks and then edit the settings to show them in different places on different pages.