You could, and it would be correct. But few would understand at first. It sounds like "caisse", like in "caisse de vin" (crate of wine).
It could also be used by an "homme de lettre" in a very pompous way.
It's much more understandable if you add "donc" after "Qu'est ce". I don't really know why, it just adds extra weirdness :)
Can you say "Quoi est-ce?" and avoid that ambiguity (and correspondingly, "Qui est-ce?")? Or would that be wrong?
You can't, the "liaison" is mandatory between "quoi" and "est".
"Qui est-ce ?" is correct. "Quoi est-ce ?" is not. "Quand est-ce" and "Où est-ce" work. "Comment est-ce ?" doesn't really work by itself but could work with something after it, like "Comment est ce film ?".
All of these work and sound more natural in spoken language if you flip and say C'est quoi ? C'est qui ? C'est quand ? etc.