This is the problem of identity, and the philosophers debate continues. If you can substitute in place, one cell at a time with a machine, as Descartes wrote, then eventually you could be replaced with a machine. Descartes believed everything then was rational and man was a machine. We lived under that idea for a while - and still do - but Godel proved it wrong. The universe is not all rational, and the Incompleteness Theorem proves it.
No, it doesn't. The Incompleteness Theorem is about mathematical logic, not about the universe, or reality. It's like saying that something cannot happen because it would violate the rules of chess (or poker, for that matter).