How to make the first widget box to be closed instead of open in admin

This is hardcoded and cannot be changed, the first sidebar box is the only one that doesn’t receives a closed CSS class.

You’d have to use jQuery to do it:

add_action( 'admin_footer-widgets.php', function(){
    ?>
    <script type="text/javascript">
    jQuery(document).ready(function($) {
        $('#widgets-right .widgets-holder-wrap:first').addClass('closed');
    });
    </script>
    <?php
});

There’s a small glitch because the box appears opened at first until the jQuery is applied. It can be smoothed with:

add_action( 'admin_head-widgets.php', function(){
    echo '<style type="text/css">#widgets-right {display:none}</style>';
}); 

And adding $('#widgets-right').show('slow'); just after the addClass.