Thanks for the tip, Massimo.

If anyone knows how to batch delete (spam) posts from an unregistered user, please let me know. Still no idea how they managed to post under "Anonymous" at all.

There is no way to post as Anonymous. When post author is removed by antispam plugins or manually wpForo shows the post author "Anonymous", it doesn't mean post has been made as "Anonymous" it means the author of this post is removed.

Any way to delete all 'Anonymous' posts?

And so far, I've encountered one post I simply cannot remove manually. See the top Chinese topic:

http://welcometotwinpeaks.com/discuss/twin-peaks-part-13/

I've deleted all caches over and over, without effect.

You can delete any topic post in Dashboard > Forums > Moderation > Unapproved or Published tab.

I recommend you install Akismet it works well with wpForo Spam Control in Dashboard > Forums > Tools admin page.

I don't see any spam protection tool on register/login form. Please install reCAPTCHA or something like this. There is no way to do a guest posting in wpForo. All your spam is made by registered users. If you protect your registration there will not be any spam. 

https://wordpress.org/plugins/search/recaptcha/