Is 320 x 240 x 16 a graphics mode that is going to be supported by the majority of graphics cards in full screen mode?
No. I don't know if it's videocards, drivers or monitors that can't do it, but that resolution is not available on lots of setups. I probably wouldn't pick 16 bit color now either. It's probably ok to be 16 bit, but I'd let B3D pick that, at least if you're doing 2d.
Is 320 x 240 x 16 a graphics mode that is going to be supported by the majority of graphics cards in full screen mode?
It's *more* sure to work, but still there are monitors which won't do anything below 1024x768 these days.
Is there another resolution that would be a pretty safe bet?
How about 640x480 fullscreen and fallback to 640x480 windowed if that mode is not available? Or is there something about your game which would make it hard to play in a window? Since 640x480 is exactly double your preferred resolution, you could probably scale that up quite nicely.
Just thinking off the top of my head, but perhaps you could do all your drawing to a texture, disable texture filtering and mipmaps and then stretch it a quad with the texture applied to fit a 640x480 screen, giving you a nice pixelly doubled resolution. Just a thought anyway.