Please don't throw around Gödel's Theorem before you've really understood it— that's one thing that makes people look like cranks!

"When does that ever happen and how does answering that question help me be more ethical?"

Very rarely; but pondering such hypotheticals has helped me to see what some of my actual moral intuitions are, once they are stripped of rationalizations (and chances to dodge the question). From that point on, I can reflect on them more effectively.

"Exposed to these situations" means to say that when someone asks about utilitarianism they say, "if there was a fat man in front of a train filled with single parents and you could push him out of the way or let the train run off a cliff what would you do?" To which my reply is, "When does that ever happen and how does answering that question help me be more ethical?"

These thought experiments aren't supposed to make you more ethical, they're supposed to help us understand our morality. If you think there are regularities in ethics- general rules that apply to multiple situations then it helps to concoct scenarios to see how those rules function. Often they're contrived because they are experiments, set up to see how the introduction of a moral principle affects our intuitions. In natural science experimental conditions usually have to be concocted as well. You don't usually find two population groups for whom everything is the same except for one variable, for example.

Digression: if a decision-theoretic model was translated into a set of axiomatic behaviors could you potentially apply Godel's Incompleteness Theorem to prove that simple and consistent is in fact too much to ask?

Agree with orthonormal. Not sure what this would mean. I don't think Godel even does that for arithmetic-- arithmetic is simple (though not trivial) and consistent, it just isn't complete. I have no idea if ethics could be a complete axiomatic system, I haven't done much on completeness beyond predicate calculus and Godel is still a little over my head.

I just mean that any simple set of principles will have to be applied inconsistently to match our intuitions. This, on moral particularism, is relevant.