Currently, the login form does not use AJAX, it works simply like the login form of WordPress https://developer.wordpress.org/reference/functions/wp_login_form/
1. You can add the login shortcode to a page and let the user stay on the same page after logging in. But I'm not sure if it can work with the modal/popup or not.
2. No, the Register button is not supported in this form.
3. The Lost Password will display under (near) the login button.
4. There is no way to integrate with 2FA authentication currently.