wp_verify_nonce keeps failing

You are not inserting the nonce field in your form, so your script won’t recieve the nonce field and this code:

 if ( !isset($_POST['nonce_name']))

Will be validated becasue $_POST['nonce_name'] is not set.

In your code, remove this line:

<input type="hidden" value="".wp_nonce_field("nonce_action','nonce_name')."'/>

And, where it said //TODO: set nonce, you need to include:

  $out .= wp_nonce_field( plugin_basename( __FILE__ ), 'nonce_name',true,false);

Note: set wp_nonce_field() ‘echo’ parameter to false to retrieve the nonce input field instead of printing it.

And then verify by:

if (!isset( $_POST['nonce_name'] ) || ! wp_verify_nonce( $_POST['nonce_name'], plugin_basename( __FILE__ ) ) )
        return;

So, your function get_form() should be:

    function get_form( $post_id=null, $tax='category' ) {

        if ( is_null($post_id) || ! taxonomy_exists($tax) )
            return false;

        $args = array( 'hide_empty' => false );
        $args = apply_filters( 'mcc_get_terms_args', $args, $post_id, $tax );
        $all_terms = get_terms( $tax, $args );

        if ( ! $all_terms )
            return false;

        $post_terms = wp_get_object_terms( $post_id, $tax, array( 'fields' => 'ids' ) );

        $permalink = get_permalink( $post_id );

        $out = "<form id='crowd-cats' action='$permalink' method='POST' >
           <ul >";

        foreach ( $all_terms as $t ) :

            $checked = in_array( $t->term_id, $post_terms) ? 'checked' : '';
            $out .= "<li>
                       <input type="checkbox" id='crowd-cat-$t->term_id' name="crowd-cat-radio[]" value="$t->term_id" $checked />
                       <label for="crowd-cat-$t->term_id" >".esc_attr($t->name)."</label>
                    </li>";

        endforeach;

        $out .= "</ul>
               <input type="submit" value="Submit" name="crowd-cats-submit"/>
               <input type="hidden" value="".esc_attr($tax)."" name="crowd-cats-tax"/>
               <input type="hidden" value="$post_id" name="crowd-cats-pid"/>";

        $out .= wp_nonce_field( plugin_basename( __FILE__ ), 'nonce_name',true,false);

        $out .= "</form>";

        return $out;

    }

And your function process_request() should be:

    function process_request(){

        // check submission
        if ( ! isset($_POST['crowd-cat-radio']) || ! is_array($_POST['crowd-cat-radio']) )
            return;

if ( !isset($_POST['nonce_name']) || !wp_verify_nonce($_POST['nonce_name'],plugin_basename( __FILE__ )) )
      {
   print 'Sorry, your nonce did not verify.';
   exit;
} else {    // continue to process form data

        // sanitize and check the input
        $suggested_terms = array_map( 'absint', $_POST['crowd-cat-radio'] );
        $post_id = absint( $_POST['crowd-cats-pid'] );
        $tax = $_POST['crowd-cats-tax'];
        if ( ! taxonomy_exists($tax) )
            return;

        // Allow only existing terms. Not sure if this is needed.
        $args = array( 'hide_empty' => false );
        $args = apply_filters( 'mcc_allowed_terms_args', $args, $post_id, $tax );
        $args['fields'] = 'ids';
        $allowed_terms = get_terms( $tax, $args );
        foreach ( $suggested_terms as $key => $term_id )
            if ( ! in_array( $term_id, $allowed_terms ) )
                unset( $suggested_terms[$key] );

        // Add terms to taxonomy
        $affected_terms = wp_set_object_terms( $post_id, $suggested_terms, $tax, false );
        update_term_cache($affected_terms);
        return $affected_terms;

    }
}