Is there any user research or testing on the smallest size a 16:9 video could play on a website, where the user doesn't feel it's too small?
This is subjective and would vary by person, age-group, and screen real estate percentage. My assessment is that no one likes to see small video - unless it was in bad quality and watching it in large (or full screen) size would make the video visually punishing.
... OR you were watching a video and working at the same time.
The MPEG-1 Code has the smallest resolution of an NTSC video stream as: 352x240. When converted from a 4:3 aspect ratio to a 16:9 aspect ratio, it becomes 352x198.