How to use this $tax_selection variable in this custom loop?

I suppose you need to make it global. It might look like:

add_action( 'pre_get_posts', 'do_user_post_selection' );
function do_user_post_selection( $query ) {

    if( !is_home() || !is_user_logged_in() )
        return;

    if( isset( $query->query_vars['post_type'] ) && ( 'nav_menu_item' == $query->query_vars['post_type'] ) )
        return; 
    if( 1 < did_action('wp') )
        return;
    global $current_user, $tax_selection;

    $tax_selection = get_user_option( 'taxonomy_selection', $current_user->ID );

    if( empty( $tax_selection ) )
        return;

    if( isset( $tax_selection['categories'] ) && ( is_array( $tax_selection['categories'] ) ) && ( !empty( $tax_selection['categories'] ) ) ) 
        $query->set( 'category__in', $tax_selection['categories'] );
    if( isset( $tax_selection['post_tag'] ) && ( is_array( $tax_selection['post_tag'] ) ) && ( !empty( $tax_selection['post_tag'] ) ) ) 
        $query->set( 'tag__in', $tax_selection['post_tag'] );

    return;
}

front-page.php

  <?php // Create and run custom loop
      global $tax_selection; // use your variable
      $custom_posts = new WP_Query();
      $custom_posts->query( array( 
          'post_type' => 'topic', 
          'posts_per_page' => 5, 
          'category__in' => $tax_selection['categories'],
      ) );
      while ($custom_posts->have_posts()) : $custom_posts->the_post();
  ?>  <article id="post-<?php the_ID(); ?>" <?php post_class(); ?>>
        <div class="entry-left">
          <h3><a href="https://wordpress.stackexchange.com/questions/77121/<?php the_permalink(); ?>" title="<?php printf( esc_attr__( 'Permalink to %s', 'twentyeleven' ), the_title_attribute( 'echo=0' ) ); ?>" rel="bookmark"><?php the_title(); ?></a></h3>
        </div><!-- .entry-header -->
      </article>
    <?php endwhile; ?>
  </div>

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