How do I pass custom shortcode-extracted variables (taxonomy) into a query function for WordPress RoyalSlider?

Two words – variable scope. Use a closure instead and pass $rstax through with use:

add_filter(
    'new_royalslider_posts_slider_query_args',
    function ( $args ) use ( $rstax ) {
        return array( 
            'post_type' =>  'carousel-slide',
            'orderby' => array(
                'menu_order' => 'ASC'
            ),
            'tax_query' => array(
                array(
                    'taxonomy' => 'carousel-slide-category',
                    'field'    => 'slug',
                    'terms'    => $rstax,
                ),
            ),
        )
    }
);