EF-S vs EF lens

You certainly can, but it's not very useful. You'll get horrible vignetting depending on the lens. Also, you will likely not be able to access the full range of focusing (macro -> infinity).

Yes, you can for most if not all of them. The best example of this is the Tokina 11-16 f/2.8. It can be mounted to a full frame camera and be shot with minimal vignetting at 16 mm f/2.8. It's quite remarkable really. However, 9 times out of 10, mounting any dedicated APS-C lens on a full-frame is a waste since you'll get horrible vignetting.