Display all posts in a custom post type, grouped by a custom taxonomy. How to sort the posts alphabetically and the terms by ID?

I think you need first to sort the genre terms in the other query:

$genres = get_terms( 'genre', 'orderby=term_id' );

And then on the posts query for each genre (like Chip Bennet’s approach):

$member_group_query = new WP_Query( array(
    'post_type' => 'member',
    'orderby' => 'name',
    'tax_query' => array(
        array(
            'taxonomy' => 'member_group',
            'field' => 'slug',
            'terms' => array( $member_group_term->slug ),
            'operator' => 'IN'
        )
    )
) );