bbPress.org

To make everything look like it’s one website is easiest to do with custom theming (or deep integration, but frankly that seems unreliable to me).

The site in question is using this plugin – https://wordpress.org/extend/plugins/wpforum/ – bbPress is not a plugin, so it cannot be integrated the same way.

You can also allow for anon posting, though I don’t know if you can make it so that people can invent their own nicks on the fly.

Can’t answer that last one, as I don’t know if anyone’s tried.