Thanks Thijs! I do appreciate it.
Great questions too. I’m not a YouTube expert, but you can customize their embed codes that are used to add a video to your site. Google has a list of the options available at https://developers.google.com/youtube/player_parameters
It does look like there is an autoplay option and another one that will let you play a list.
There’s also a plugin called YouTube Embed Plus (http://wordpress.org/plugins/youtube-embed-plus/) which indicates it will provide an easy way to set things like autoplay. That might be the easiest option.
WordPress does have a built-in video player as of version 3.6, but I don’t believe it has an autoplay feature so if you want to do that, I would place the video on YouTube and make use of their features.
For the random option, that certainly could be created. The way to go about that without having to create that from scratch would probably be to use a slider plugin. You’ll need one that supports videos, where you can disable auto-advance and also has a random slide option. I’m afraid I don’t have a recommendation there as to which one to use, but if you can find one that can do those things, you should be in good shape. The Soliloquy slider is often recommended, but I personally haven’t tried it (http://soliloquywp.com).
If I can help out, just let me know.
Thanks again,
Bill