Does the header have to be on every page?

depending on what you define as ‘header’ you can use a conditional statement in header.php to show only part of it on the front page;

https://codex.www.ads-software.com/Conditional_Tags

before starting with customisations, create a child theme to work with – do not edit the default theme Twenty Twelve directly.
https://codex.www.ads-software.com/Child_Themes

Excellent, thanks “is_front_page()” is what I need.

Already using child theme (the plug-in “One Click Child Theme” is fantastic)