Recreating the hierarchy of taxonomies for a dropdown form menu?

You can use the standard WordPress function, get the dropdown already formatted and solve both problems at once.

Like so:

define( 'WP_USE_THEMES', false );
require( './wp-load.php' );

wp_dropdown_categories(
    array(
        'child_of' => 0,
        'class' => 'postform', 
        'depth' => 0,
        'echo' => 1,
        'exclude' => '', 
        'hide_empty' => false, 
        'hide_if_empty' => false,
        'hierarchical' => true,
        'id' => '',
        'name' => 'cat-dropdown', 
        'order' => 'ASC',
        'orderby' => 'name', 
        'selected' => 0, 
        'show_count' => 0,
        'show_option_all' => '', 
        'show_option_none' => __('None'),
        'tab_index' => 0, 
        'taxonomy' => 'category',
    )
);

Outputs:

<select name="cat-dropdown" id='cat-dropdown' class="postform" >
    <option value="-1">None</option>
    <option class="level-0" value="2">Other</option>
    <option class="level-0" value="1">Uncategorized</option>
    <option class="level-1" value="4">&nbsp;&nbsp;&nbsp;Other</option>
    <option class="level-2" value="5">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Other</option>
</select>

For the default category: