Hi,
– Thin Gray Lines
– Mobile Menu Empty

Please add the following code in:
WP-Admin Dashboard > Appearance > Customize > General > Custom Codes > Custom CSS

.cl-row .bg-layer{
  background-color: #000 !important;
}

.cl-mobile-menu nav>ul>li{
  opacity: 1 !important;
}

– Increase the Sidebar width
The width of the header/sidebar on the left is preferred size for this type of layout. However, you can change the sidebar width by adding the following code in the customizer(path is mentioned above)

@media (min-width: 992px)
 .header_container {
    width: 285px; /* Change width of sidebar */
 }
}

– Only two social icons display on the Mobile device
There’re only two social icons displayed because for all social icons the layout will be messy and condensed. The logic to display the first two social icons is to make sure the layout of the header should be clear and focused.

– Add links to social icons

You can add links to social buttons simply through
WP-Admin Dashboard > Appearance > Customize > Hover over the social icons > Click on SOCIAL ICONS > Add/Remove links and social buttons > Publish > Done.

– Empty Space below the logo

The space below the logo depends upon the content below it. The more the menu items the lesser will be space below and above the menu.

Thanks,

Hi,
Add the following code in the customizer

.cl-mobile-menu nav>ul>li{
  opacity: 1 !important;
}

Thanks,

Hi,
– Menu not appearing
Please add the following code:
WP-Admin Dashboard > Appearance > Customize > General > Custom Codes > Custom CSS

.cl-mobile-menu nav > ul > li{opacity: 1 !important}

– Font issues
Can you enlist the steps you’re following?

Thanks,