Is it possible to choose a video in the Email Editor from a video uploaded to the Design Studio, or does the video need to be hosted by YouTube or Vimeo?
It needs to be hosted.
The Email Editor video function is just a quick way of adding a triangle/play symbol overlay on top of a thumbnail of the image. (PS., It doesn't work in all email clients. )
The actual streaming of the video is usually best achieved on YouTube/Vimeo/etc..
Whilst you could upload a video file to the Design Studio, it wouldn't be useful in this context.