If I use Windows Media Player to send a video to a projector, I start with the Windows folder on the screen, click on Full Screen and Play, then move the cursor off to one side and wait for the graphics at the bottom of the screen to clear so the video can play on. This is none too professional.

So is there software out there which would let me start with a black screen with the video cued and waiting, and launch it with a hot key with nothing but the video showing on the projector?

An embedded player in Powerpoint will let you do this. IMO you're looking for presentation software...