Multiple domains with different child themes on one installation?

Use the plugin WordPress MU Domain Mapping. Read the installation instructions very carefully. There is also an ebook WordPress Multisite 101 where you can read more (like migration from single site installation etc.).

You can activate themes per single site as usual. Or set the constant WP_DEFAULT_THEME in your wp-config.php depending on the host name:

switch ( $_SERVER['HTTP_HOST'] )
{
    case 'example.com':
        define( 'WP_DEFAULT_THEME', 'blue-child' );
        break;
    case 'example.net':
        define( 'WP_DEFAULT_THEME', 'red-child' );
        break;
    default:
        define( 'WP_DEFAULT_THEME', 'parent-theme-slug' );
}