How to show featured post first, then separate loop for other posts

You’ve got a whole lot of problems here. First you’re asking for Tag_slug__in, but providing a title, secondly tag_slug__in accepts an array (I’m not sure it will work with a string). Both issues will cause problems because titles are not slugs and arrays are not strings.. Maybe just use “tag”? But what if page title doesn’t equal a slug? Then you have an error and no error message to say what happened. (which i believe is what is happening here.) Or you have more than one featured post?

You have 2 custom queries but you’re not reseting post data in either of them.

I asked to see the rest of your code, because I don’t know what kind of page this is on to relate to the “get_the_title()” reference in your args . Is this is an archive, template, category, or taxonomy page etc.

Here is a code sample to try. I’ve commented out your tag_slug__in from both your queries. I’ve added post resets, I’ve added an actual return of content for you in each.

<?php 

// Attempting to show featured first: the array pulls one post from the 'Featured' category with the relevant tag. This works correctly

$my_query = new WP_Query(array(
            'category_name' => 'featured',
            'posts_per_page' => 1,
            //'tag_slug__in' => get_the_title() 
            ) 
            );

while ( $my_query->have_posts() ) : $my_query->the_post();

$do_not_duplicate = $post->ID; ?>

<!-- Code to display featured post with relevant formatting goes here -->
echo 'My Featured Post: '. get_the_title().'<br><hr><br>';
<?php

// This ends the first loop

endwhile; 
wp_reset_postdata();?>
<?php 

// The query for standard posts: this is supposed to check posts for the relevant tag. This loop worked correctly before I added the loop for featured posts

$the_query = new WP_Query(array(
            //'tag_slug__in' => get_the_title(), 
            'posts_per_page' => -1 
            ) 
            );

// The loop to display these posts according to the query

if( $the_query->have_posts() ):

    while( $the_query->have_posts() ): $the_query->the_post(); 

// This line is from the Codex, and is supposed to prevent duplication of the featured post. I'm unsure if this is implemented correctly?

    if ( $post->ID == $do_not_duplicate ) continue; ?>

<!-- Formatting for regular posts -->
echo 'Not a Featured Post: '. get_the_title().'<br>';
<?php

    // Ending second loop

    endwhile;
wp_reset_postdata();
    endif;
?>