First, my server is located in The Netherlands, as far as I know only the law applies of the country where the site is being hosted.
Second, our RIAA (Buma/Stemra) made a deal with Youtube that ensures every Dutch citizen can embed YT videos without any problems (see: http://www.bumastemra.nl/en-US/OverBumaStemra/Actueel/Buma_a... and this from their FAQ http://i55.tinypic.com/svi1jt.jpg). It says that I don't have any problems as long as I don't monetize it. Since I hate banners and my costs are really low this ain't a problem.
Third, I actually don't store _any_ data or playlists. What I've done is coding a few scripts that let you crawl Youtube and Last.fm in a smart way. In that POV i'm not different to a regular search engine with a 'share your results' button, futhermore my server does totally nothing after you received the javascripts, it's just your browser and the other sites.
Last but not least, it's impossible for me to measure how many and what kind of videos are being played. All this stuff works with # anchor urls, my server simply can't register that.
I'm very curious in what other people think about this, can you ask your profs?