Barbie's full name is Barbara Millicent Roberts. However, Ken's last name has not been officially established by Mattel, the company that produces Barbie dolls.
Regarding their relationship status, Barbie and Ken are often depicted as a couple, but they are not officially married. They are portrayed as dating or being in a romantic relationship. However, keep in mind that Mattel's storytelling and character backgrounds may evolve over time, so it's possible that there have been updates or changes.