downloading the actual YouTube video (possible? legal?)
It is possible and legal and this happens every time you watch a video. You may have problems to locate it in the local browser cache. Tools like DownloadHelper - Media download Firefox extension allow to select the format (240p-1080p if available) and save the file into a random folder.
Did you read Terms of Service - YouTube ?
You agree not to distribute in any medium any part of the Service or the Content without YouTube's prior written authorization, unless YouTube makes available the means for such distribution through functionality offered by the Service (such as the Embeddable Player).
Having the authorization you may also get a tool to access the videos more easy or do this manually. The videos may also contain Sony, BMG, ... content which is likely copyright protected, so one needs to review them manually. Developing and maintaining an application to do this may be more expensive.
We've been having some internal discussions lately on how to best work with the "great firewall", but the concerns I've heard have mainly been around degraded performance and page load times. How are things in that regard? A page that takes 20 seconds to load can be a deterrent as well. Let me float this internally and see if any of our network folks have an idea.
For video, have you explored any third-party video modules to host the content instead of YouTube, either our own or Brightcove or Vbrick or the like? That would get you around the terms of service issue, anyway.