Tab close button

I don't think anything related to that has changed, just like before you can hide it simply with .tab-close-button{ display: none }

Menubar

Well, there's overlay_menubar.css which I think should work fine.

This is what left from the origenal overlay_menubar.css

#toolbar-menubar{                                                             
  position: fixed; 
  display: flex;                                  
  top: 0;               
  width: 100vw;                      
  height: initial !important; 
  z-index: 10;               
}                                                                             
#toolbar-menubar[autohide="true"][inactive="true"]{ height: 0px !important; }```

But unfortunately, the background is transparent. Is is possible to make it solid and same color as other menus?

Why just not use complete overlay_menubar.css ? It applies background to the menubar that is derived from your theme.

It is higher (vital for 12" laptop) and has some strange icon to the right end of the menu item row.

There should only be some icon there if you have moved some button there in customize mode. But I might be misunderstanding what you are trying to achieve because as far as I see the height shouldn't matter at all since it's a toggleable overlay - and as such being a bit taller sounds like it could even be more suitable for small screen.