Hi Jonathan!
Below is a screen shot of the Element Inspector Panel with the various options for video - one of which is whether or not to display the video controls.
As for the other part of your question the video itself can have alpha channels - but I am guessing You are referring to having the actual video background be transparent. If this is what You mean the short answer is not at this time. This capability (or lack there of) is browser, not Hype, related.
An image sequence of transparent PNG files would be a work around. Unfortunately, Hype does not automate the import of image sequences - i.e. bringing them in a linear fashion and spec the frame rate. An animated GIF would be another option - but it is rather weak tea compared to what a video or animated PNGs could offer.
The following is a further discussion of transparent background for videos: