Do you think it's safe for a young woman, age 21, to travel throughout Europe alone?

No! You have reason to worry. Even if she's an adult, it's just not safe to travel alone to other countries by herself. It this age and era, it's just too, dangerous. People have done so, but they have had ways of communication like email and a cellphone(can't believe she doesn't have a cellphone:)). Plus, it kind of sounds like she doesn't have a lot of information about this trip. If she goes, make sure she has a cellphone!