Hello,
1) We use the default WooCommerce function.
2) I don’t remember which third-party plugins customers used for the Ajax filtering, truly saying. You may try one of the free plugins https://wordpress.org/plugins/woocommerce-ajax-filters/ or https://wordpress.org/plugins/yith-woocommerce-ajax-navigation/ but I suppose you need to use correct selectors in the settings of the plugin in any case.
Regards