Top menu disapperad

[=] icon needs color change, becuase by default in that template was white on dark background as you could see in demo.

Give me 90sec...
DONE

I suggest to:

1. Add Search as menu item, you can also use Icon instead of "Search" Name. Watch my YouTube video about that topic > https://www.youtube.com/watch?v=iJL3rSqVb1U
2. In Joomla 4 and Joomla 5 you should use only Smart Search (Finder) - but it must be enabled and content must be indexed from component settings. Read more in Joomla offcial documenation and watch Youtube clips.
3. Smart Search plugins (for used components) must be enabled as well.
4. In Mobile Menu you can always publish a Smart Search module. Just use "offcanvas-modules" position for that. And it will be displayed at the bottom, after menu items. But having a Search Mneu item is also recommended way of doing that.

ad 2) Helix Ultimate as may know uses Bootstrap 5 - it means that you can use all class names from it.

ad 1) At least on Browser Mobile emulator all is visible (Firefox/Chrome)

Probably Search bar is too wide for very small screen, and has to be reduced with max-width.

I thought you will show me screenshot from real Phone.

Only via Custom CSS. This should help

@media (max-width: 840px) {
  #sp-logo.has-border {padding-right: 0;}
    #sp-menu {padding-left: 0;}
}

even without reducing width of search field. Padding less between make a big difference.

Have you refreshed browser 2x to reload CSS.

You can add also that line

@media (max-width: 580px) {
  #sp-header .sp-module {margin-left: 0;}
}

Is OK if you turn phone to horizontal view?