There are only a few thousands of Karaites, who almost all live in Israel. They should not be confused with Christian heretics who CLAIM to be Karaites but have no historic connection.
Lots of web information about real Karaites on the web, as long as Christian interpretations are noted.
As with anyone who is thought to be or claims to be Jewish, the official determination (such as for marriage) is made on a case-by-case basis.
Interesting -- I had roughly heard of Crimean Karaites, but your post indicated more details. These early medieval communities like Karaites, Mandeans etc. are fascinating historically as remnants of earlier populations or religious thoughts