JW Player is a very excellent HTML 5 video player that should cover most (all?) of your needs. I've used this on many websites and I can honestly say that it is one of the best players on the market. They have a free version as well, you just need a license for commercial use.
I would suggest MediaElement.js as well, but I don't think it does playlists, though it would be quite trivial to build that functionality (if someone else hasn't already).
Someone has also kindly put together a comparison list of HTML 5 video players for your perusal