Of course - that's the current standard way to produce HDR photos. The advantage here is being able to do it all in a single image, which prevents mismatches between images due to subject or photographer movement.
Yes, but if I understand correctly, unless those images occur at the same time (or really, really close) you run into issues involving subject and camera motion.
Because the images would look funny. Imagine you're capturing someone running - at frame 1, they would be at position x, and at frame 2, they'll be at position x + 1. If you try to stack them together, you'll get a weird ghosting effect.
Motion blur, but different parts of the image would be exposed differently due to the nature of HDR, resulting in an odd look. Imagine instead of a runner, that you're panning from a dark to light scene.