I was able to solve my problem.
My issues connecting to a projector were actually pretty severe. It started with just my VGA/DisplayPort connector not working, so I switched to HDMI; then HDMI would not work; then I started to get a black screen with the cursor only, and discovered that if I blindly clicked on a menu, stuff would appear, but only in a very disturbing way.
Eventually, any time I plugged in either connector, my Retina MBP's resolution would seem to switch, but then the screen would go blue, and nothing would work--the machine froze and had to be force-shut down and restarted.
My solution:
I installed Display Menu, a free utility on the App Store:
http://itunes.apple.com/us/app/id549083868
Before plugging in my HDMI, I set the display resolution to a non-HiDPI setting (e.g., 1680 x 1050. For some reason, doing that with this utility made it work for me. My resolution went to Full HD (1920 x 1080, IIRC) and everything worked OK, the projector was recognized and displayed my screen. I have not tried the VGA adaptor or tried to turn off mirroring, though. Didn't need to and didn't want to deal with any other hassles...