How to Grab first Image from some Chosen Categories

Question

Your solution seems overly complicated to me. WP_Query can do nearly all of the work, you just need to construct a meta_query to look for thumbnails. The code is slightly truncated but here is the idea:

$categories = array(1,2,3); // your specific category IDs
$first_thumb = new WP_Query(
  array(
    'posts_per_page' => count($categories),
    'category__in' => $categories,
    'meta_query' => array(
      array(
        'meta_key' => '_thumbnail_id',
        'compare' => 'EXISTS'
      )
    ),
  )
);
if ($first_thumb->have_posts()) {
  while ($first_thumb->have_posts()) {
    $first_thumb->the_post();
    the_post_thumbnail();
  }
}