The footer sections are just widgets. You can add the search widget where desired in the footer.
Admin Dashboard>Appearance>Widgets>Drag the Search Widget to the Footer section you want it to appear in – See:
https://codex.www.ads-software.com/WordPress_Widgets#Existing_Widgets_in_Existing_Widget_Areas
Removing the search from the menu area in the header can be achieved per this post:
https://www.ads-software.com/support/topic/how-to-remove-search-bar-from-header?replies=2
Additional note: Please review why using a Child Theme is important at:
https://codex.www.ads-software.com/Child_Themes
As a workaround, you can add this CSS:
.menu form {display:none;}
To discover what CSS is output by your theme, use a web inspection tool such as Firebug: https://getfirebug.com/ , Chrome Inspect Element: https://developers.google.com/chrome-developer-tools/ or Internet Explorer 9+ F12 Dev. Tools: https://msdn.microsoft.com/en-us/library/dd565627(v=vs.85).aspx#csstool
There are others.
When editing CSS, use a Child Theme
https://codex.www.ads-software.com/Child_Themes
Custom CSS Plugin, or Theme provided custom CSS option.
Edits to parent themes are lost on theme update.
Learn CSS: https://www.w3schools.com/css/