Remove the "header" section from pages

In the Theme Options section, there is a "Header" tab that is directly related to some text that goes under the logo/navigation area. It allows for a Header Title and a Description that is smaller text. I have two questions concerning this area.

1. The button that is right justified. This "action link" displays even if there is no text entered into the field. It defaults to "Add your link." I'm using a child theme, but can someone help me figure out what file needs to be copied from the parent and how it should be edited to remove this button?

2. This Header section - if turned on - is displayed on every page of the site. If I just want this Header section to be on the home page, what template(s) need to be copied over to the child and how must they be updated?

Thank you!