Videos are not stored in every server. Nobody would have been able to pay for the bills if that was the case.

The videos and images stay on the origin, and are fetched from the origin.

Afaik admins that enable the image proxy cache only the images, not videos.

This depends on the instance of the video being used. Some instances clone a copy of the video, some use proxies, and some send the link directly to the user. I don't recommend it if you have limited bandwidth.