Both sentences are correct, and there is not much practical difference between them. The second one contracts "would" with "not". That affects the word order, but that is perfectly normal for such contractions. Contractions tend to be found to a greater extent in casual writing, colloquial speech, etc. Other than that, I see no major difference between the sentences.
There is a (somewhat minor) difference in pronunciation. In the first sentence, you could stress "not", but in the second, you couldn't stress the clitic "n't".
I'm a bit confused about your last question: "Is it correct to say the first sentence puts an emphasis on the negation, meaning that it's a bit shocking that we do not have that expectation?" Your sentences seem to indicate that we DO have that expectation.