T
Check navigation - items - system - toolbar site,
and if messenger is hidden for phone or other exclutions. You can set a smaller icon here. For changing viewport for small devices you can modify main css in "custom styles" in the settings of your template.
If you want to turn search completely off, just deactivate it in Toolbar Site menu.
I did that in ...path-to-site/studio/builder_menu.php?page=items&module=system&set=sys_toolbar_site but search icon still present....
Try to clear cache.
