How do I nest conditionals to identify sub-site & other criteria?

You should use get_current_blog_id() if you need just the id and get_blog_details() if you need more fields.

Be aware that get_current_site in fact gives you information about the network of sites in a WP_Network object. Yeah, the name is kind of confusing, as noted in the Codex but not yet on developer docs:

Note: get_current_site() was inherited from MU, and is named based on the old terminology which spoke of multiple “blogs” on a “site”. We now speak of multiple “sites” on a “network” instead, but some of the old terminology still lives on in some function names. This function returns information about the current network. To get information about the current site on the network, see get_current_blog_id() and get_blog_details().

It would be something like this.

// get current blog id
$blog_id = get_current_blog_id();
if ( 2 === $blog_id){
    // if site id is 2
    //middle part for this case
} else {
    //if is not
    //middle part for this case
}