.gov is managed by the US government in the exact way you describe. There is nothing "inherently international" about it. It isn't meant for anyone outside of US government agencies.
Plenty of exceptions abound, though: https://en.wikipedia.org/wiki/.gov#Use
Come to find out that the k12.ca.us. domain is completely defunct! There is no SOA for it, and WHOIS indicates it's inactive. k12.ny.us. seems to be in the same condition. However, there are still k12.<state>.us. domains in other parts of these USA. If you search for them, you can find websites and email addresses under that hierarchy.