How much of the Hyborian age is under public domain, and how much of it is copyrighted? Is it similar to the status of the Cuthulu Mythos in any particular way? For example, would there be any legal barriers that would prevent me from writing a story about a Pict-Cimmerian blood feud (that doesn’t involve Conan)?
My thoughts, with respect to the U.S.: My understanding is that some of the Conan stories are in the public domain, and that REH's essay on the Hyborian Age is as well. To the extent there are stories set in the Hyborian Age that are still under copyright, it could be a potential problem if you use those story elements. If it's true that the Hyborian Age essay is in the public domain, using information drawn from that is fine. If you're going to market the work using the phrase "Hyborian Age" you would want to look into any potential trademark issues. There is a trademark registration for "The Cimmerian," but that wouldn't necessarily prevent you from having a Cimmerian in one of your stories if you're drawing on public domain works or your own contributions for the Cimmerian.
I doubt there would be copyright issues with it. Also all of the peoples involved, including Picts and Cimmerians, are historic. REH wasn't overly concerned with historical accuracy but he was mining current popular anthropology and history as well as occult theories from theosophy and elsewhere.
To the extent one is relying on historical details of Cimmerians, no issue. REH fictionalized his version, and there's a possibility of an issue to the extent one uses his version of Cimmerians and the details are from works that remain under copyright.