Problem:
The issue here is that the user is getting the message "cannot connect to toolset dot com"

Solution:
This error generally occurs when you've migrated your site and the url changes and your Toolset plugins can no longer validate with our servers.

To check if your plugins are able to validate correctly you will need to go to Toolset -> Dashboard, then click on help in the top right hand corner. Select the Toolset Debug tab then click "Toolset Debug Information and Troubleshooting"

From there you will be taken to the Toolset Debug information. Scroll down on the page until you see the link for "Installer Support". Click this link.

You should now be able to see your site validation information. If you're see "Toolset site key not matching for http://www.xyz.com" then you will need to regenerate your site as the URL may have changed and your key isn't validating for the current URL.

The best way to generate the site key is by following the instructions in the link below.
https://toolset.com/faq/how-to-install-and-register-toolset/#automatic-installation-once-you-have-types-plugin-installed

Paying close attention to step 3.

Once you've done this go to Dashboard -> Updates and click Check Again and the warning notice should no longer appear.

Problem:
Add custom user roles to author quick edit dropdown.

Solution:

This can be done by adding the following code to your functions.php file

add_filter('wp_dropdown_users_args', 'assign_any_users_author_box_func', 10, 2);
function assign_any_users_author_box_func($query_args, $r){
    $query_arg['who'] = 'any';
    return $query_arg;
}

It should be noted that the user role must also have permissions for the post type that you want them to show up on.