Nameservers
Every hostname you mint is answered authoritatively by nameservers we operate. There is nothing for you to run, configure or keep patched.
They answer from the United States (ns1.mintns.com) and from Europe (ns2.mintns.com) today, and we are adding locations. A domain on a paid plan is served from Google's Cloud DNS or AWS's Route 53 anycast networks, so a query is answered from wherever it was asked.
The nameservers are their own system, separate from this website and from the API. Creating a hostname goes through the API; answering for one does not.
You can see their status, and how often they have answered over the last ninety days, on the status page. The standards each one implements are on standards.