Moving the WordPress dashboard avatar over the admin menu

You could make a custom menu item using a plugin called Admin Menu Editor

And you could hide the avatar using CSS, to do this you could use a plugin to insert CSS in the admin dashboard or just create your own admin.css file and insert it putting this in your functions.php file:

function registerCustomAdminCss(){
$src = "https://wordpress.stackexchange.com/questions/293835/path/admin.css";
$handle = "customAdminCss";
wp_register_script($handle, $src);
wp_enqueue_style($handle, $src, array(), false, false);
    }
    add_action('admin_head', 'registerCustomAdminCss');

Where path/admin.css is the path to your admin.css file