I'm not a logician by a long shot, so I probably can't explain that correctly. I think Gödel found a way to make a logical proposition refer to itself, and then found a way to assert provability. He could then construct the sentence "this sentence is not provable". He showed that such a sentence must exists within any system of sufficient power. Thus the system must be self-contradictory (inconsistent), or the sentence must be true (and the system must be incomplete). I'm not sure if such a sentence still refers to itself when negated, so I can't answer the last one.
My understanding of the incompleteness theorem is that, for a given set of axioms, there will be unprovably true things. Changing the axioms would change which things were unprovable.
