uri

That’s an interesting setup, I will definitely look into that for the next update, However, the way things work now that’s probably not possible.

You can embed shortcodes within the enclosing membersonly shortcode, if the thing you want to be members only can be a shortcode you could then use shortcode widget in elementor https://elementor.com/help/shortcode-widget.

 [membersonly display="Login To Download" linkto="/wp-admin/" linktext="Login Here"]

 protected content here [Gallery]

[/membersonly] 

Not a 100% sure but I think that’s the default behaviour for Multisite, maybe you can get around this problem with a frontend profile editor plugin, you will need one that supports Multisite.

I’m not sure exactly where the issue is but it seems to start here:

//Adding Javascript
add_action( 'wp_head', function () { ?>
	<script>

You can try the following, be sure to do this on the local machine or a staging environment or do a Back-Up beforehand

//Add the Search Form and Javascript
//Set theme_location parameter by the menu locations theme offers - Is it in Primary menu? 
add_filter('wp_nav_menu_items', 'add_search_form', 10, 2);

// Display fontawesome search icon in menus and toggle search form 
function add_search_form($items, $args) {
if( $args->theme_location == 'primary' )
       $items .= '<li class="search"><a class="search_icon"><i class="fa fa-search"></i></a><div style="display:none;" class="spicewpsearchform">'. get_search_form(false) .'</div></li>';
       return $items;
}
//Adding Javascript
add_action( 'wp_head', 'addjs_header_code');

// start php to add Javascript
function addjs_header_code(){ ?><script>
// Display fontawesome search icon in menus and toggle search form 
function add_search_form($items, $args) {
  if( $args->theme_location == 'primary' )
         $items .= '<li class="search"><a class="search_icon"><i class="fa fa-search"></i></a><div style="display:none;" class="spicewpsearchform">'. get_search_form(false) .'</div></li>';
         return $items;
  }
    </script><?php 
} // end of php

This sounds like WordPress Multisite can help here.
You have the ability to create a network of sites by using the multisite feature.
multisite feature

I cant was not able to help with all your questions but I can give some feedback

2. On the HQ landing page, when the user clicks the link to go to a location’s site, add some onclick JS in the a tag to set the “location” cookie to that location with the jQuery.cookie JavaScript function (not sure if WordPress has jQuery installed by default)

Yes WordPress includes jQuery here is more on how you can use it:

• https://developer.www.ads-software.com/reference/functions/wp_enqueue_script/
• https://stackoverflow.com/questions/16823679/wordpress-plugin-development-how-to-use-jquery-javascript