YouTube video assignments?
Should it be publicly or privately posted on YouTube? From my
  understanding, the videos are being made public.


Private, and the instructor should be telling students how to do this and reprimanding them if they are not. 


  What advantages with respect to simply asking for the paper to be
  handed physically or scanned and e-mailed?


well, the student seems to be explaining their thought process in the video. this could help the professor correct students who are astray. Something that will be hard to do when students are remote.


  Does it provide good proof of authorship? I.e. people can hardly
  re-use a handwritten paper delivered to the teacher, but knowing that
  the student's face is not on the video and the teacher may not
  recognize the student's voice, I'm unsure it gives reasonable
  confidence that the student is behind the microphone doing the work.
  Maybe a student reposts an old (deleted) video if the exercise has
  been used in the past.


This seems like it could be an issue, but I feel as though it would be about as big a problem as other forms of cheating with "in person" assignments. If teachers can recognize two similar papers, there is a pretty high chance that they would remember an entirely copied video. You also should be able to notice if someone isn't able to explain the work they did properly.


  It this nice for one-shot experiences, or is it functional for
  continued usage? I.e. could a teacher give the same video assignment
  over several years? Possibly with the same exercise problems?


Functional for continued use, although the instructor should still be teaching each week. I had an instructor record each lesson and upload it to a private YouTube playlist for review. If class was missed he would simply take a previous years video and upload it, and then ask if there were any questions. It was insanely helpful as a student and I think almost every teacher should do it (I still go back to some of those lectures for review and the class was over a year ago).


  Are there any intelectual property or image rights concern? I.e. for
  music/movies, performers/actors are continuously paid for exhibitions
  of their performances, while intellectual properties such as the
  contents of one's thesis are often waived to the university. What if
  these videos are later included for paid online courses?


Yes there are, but if the videos are private and only used by students in the class, you should be fine. Videos like this should only accompany an in person class, different things should be done for a completely online class. 


  Is there any advantage for the professor with respect to simply asking
  the video file to be delivered on a Dropbox folder or similar cloud
  storage method?


YouTube is probably easier for students but Dropbox is probably a little more secure. By that I mean the only security private YouTube videos have is that you can only find them if you have the proper link. In theory students could share the link and work ahead, or someone outside the university could have access to the class. I have known some teachers that won't record lectures because of this (god forbid learning takes place).