Slavakion

On second thought, consciousness isn't really the right word. But I'm not sure what is. Consciousness is your awareness, and what you're talking about I can only describe (lamely) as animation. Since they're closely related, I'll try to answer your questions from both points of view. Keep in mind that I'm not a theologian, psychologist or biologist.

Consciousness is maintained by the human body in order to preserve life. Animation, on our level, is the byproduct of coexisting cellular structures trying to keep themselves alive. Animation on the cellular level is maintained by the interplay of organelles. Animation below that...?

After the support system dies... is the person on artificial life support? If so, the human body will tick along just fine. If not, animation and consciousness will cease.

Without consciousness... To what level? If someone is unconscious, for example because of too much alcohol, life can continue. If someone is permanently no longer aware of their surroundings, perhaps not. If you are completely unaware of what is happening, then how would you know that you need to breathe, or pump blood, etc. If animation ceases, there is nothing to keep you alive. Animation is what causes the peculiar things to happen that cause life. The only thing is that somebody can be clinically dead, devoid of animation, and be brought back to life. Reanimated, so to speak. Hmm...

Consciousness should return if somebody passes out. If it involves brain damage, maybe not. Where does it go? It's nothing physical, so it can't go anywhere. Consciousness is... caused by electrochemical signals racing through the brain to produce something wonderful. Animation is the same way.

If you are religious, then you could much more easily sum this up by saying that animation is caused by your soul, which floats out of your body after death.