PhDs in the arts - what exactly do they do?
In short: yes, and a whole lot more.

A PhD in the arts usually most closely resembles other PhDs in the humanities, like English or History, but often have a more experiential/practical component as well. Depending on the program and field of study, it might be somewhat of a mash-up of psychology, anthropology, physiology, literature, history, political science...whatever relates to one's area of research and study!

I'll give a concrete, real-world example: in the Psychology department of my present university there are faculty and students (undergraduates and those preparing for masters/PhD studies) conducting cross-departmental research with the Theatre &amp; Dance department on the "Psychology of Dance". In one experimental study they looked at inter-rater reliability of those who are not particularly trained or experienced in interpretive dance, and then showed them specially created dance performance videos. They were looking to see if interpretive dance really is a genuine form of communication: can the artists/performers communicate specific emotions consistently to the audience such as anger, fear, affection, companionship, etc? Or is it all just people assigning arbitrary meanings, like people giving serious meaning to a randomly generated drawing? Their results so far indicate that especially for some emotions, there is a very strong inter-rater reliability: even those who have never studied/seen interpretive dance before can identify certain emotions, activities, representations, etc - and as usual they work on writing up and publishing such results, giving talks, etc.

As another example, at NYU they have a Performance Studies program and they have a lot of detail on their website about the kind of things student's are doing. These include things like studying different styles of dance, dance in other cultures, documenting and studying modern performance art, and so forth.

As with most PhDs, the purpose of them is research and instruction, but what one does with it varies. Some will be instructors, professors, researchers, owners of studios, contributors in community organizations, lecturers, writers, performance artists...some will go into 'industry' as choreographers, directors, producers, and so on. It's a far cry from Underwater Basket Weaving!

One thing I'll note from personal observation: many, many people working in the arts choose not to go by the title "Doctor" or even "Professor". This seemed odd to me, but it's a culture thing - they just don't do it. So, many people who have a PhD in the arts, especially in music and dance, don't advertise their education in any way - so you've probably known more people with a PhD in the arts than you realize!