I have had a privilege of knowing what concerns the admission committee about my applicant's file for pure math degree. As of right now, the most significant concern expressed was my weak performance on the GRE subject test (in math.) My question is, would the committee care about the perfect performance on AP Calculus tests, AB and BC? Could it somehow help to mitigate the less-than-appealing GRE math score?
I would say a graduate admission committee in pure mathematics would consider AP Calculus scores completely irrelevant.
Good scores (even perfect scores) on these exams show only that you are (or were) competent at high school / freshman calculus; if this were not the case, you probably should not be applying for grad school in mathematics at all. As a mathematician would say, calculus competence is necessary but not sufficient for success in a grad program. Of more direct relevance is your preparation in more advanced, proof-based subjects (abstract algebra, real and complex analysis, etc), of which the GRE is a better (thought not necessarily good) measure. You must have observed that these courses are of a very different flavor from calculus, with much more theory and less rote computation; at the graduate level this trend will only continue.
Even if you are brilliant at calculus, math professors know that this does not necessarily carry over into more advanced subjects. They really want to see what you have been doing lately.
(Disclaimer: I am a mathematician but have not been on an admission committee.)
No. Half the people who take the Calculus BC exam get perfect scores, so you can assume almost every applicant has a 5 on the BC exam. (Except they wouldn't say so, because by the time they graduate college they usually have other accomplishments that carry more weight.)