Please try to activate parent theme on site (Appearance > Themes) and check if problems persist. Also, temporarily disable all 3rd-party plugins to be sure that no one of them is a reason for problems. Let me know results.
On my test site, the search and menu work fine, so to fix it on your site we need to find out what caused problems.
P.S. Your settings won’t be lost after changing the active theme on site, but you can export theme options to file via Theme Options > Import/Export > Import or Export your theme configuration. Some widgets may change positions, so I would recommend you check where every widget is placed in Appearance > Widgets or you may open widgets in a new browser tab, before the parent theme activation, to be able to compare widgets position after changing the active theme to child theme back.