I was recently looking for a simple, robust, usable, non-branded swf video player to use as a flash fallback when using the HTML5
element, and I found the Strobe Media Playback player from Adobe. It fills all those requirements, and is fairly feature-rich. The files are directly downloadable from their Sourceforge directory. Here, more or less, is how I use it:
It supports lots of parameters specifiable from within the flashvars
parameter, such as autoplay
and a whole host of parameters related to the controls (the documentation lists them).