Yes, they are accepted on the application. Many of the undergrads at MIT qualified for the AIME. A substantial number qualified for the USAMO or IMO as well - just look at the Putnam results every year.

Frankly, in order to stand out among MIT applicants by demonstrating some sort of mathematical ability on these exams, the minimum is probably USAMO qualification (top 270 of ~200k AMC takers). Otherwise it's a nice thing to have but not particularly unique. Even then I know a reasonable number of USAMO qualifiers who have been rejected.

Source: was a MIT undergrad < 5 years ago

Yes. But it is optional.