I've used freedns.afraid.org.

You can run up to 5 domains on there free of charge. On their about-us page, they state that all the money generated from premium users goes back into the company. Their domains are all public, and are "donated" (not sure if that's the word) by many generous people. Their site says they're currently getting around 7 billion dns requests a month.

The website was started by a hobbiest developer who wanted "to create a safe environment where other programmers could share domain names with one another at no cost".

I'm not sure that it really counts as FOSS, but it provides a way for anyone to use a domain free of charge without having to own any server infrastructure themselves.

https://freedns.afraid.org/about-us/

Edit: also just realized this post is 3 years old, sorry for being a bit late to the party! I hope this helps someone else :)

No. Privacy alone is hard and tricky. ICANN doesn't really allow privacy. Though NICs located within the EU protect your privacy since they have to comply with GDPR. ICANN tolerates that they break their rules and works on rew rules that allow to comply with GDPR. ICANN requires your mail address, phone number and email address to be publicly available via whois. You don't offially own the domain when a registrar offers private registration where they put there their own contact data.