During the U.S. admissions process to PhD programs in the STEM fields, are the admissions committees typically notified by university administrators, if one of the applicant's family is a generous donor to the school?
If the answer is yes, does this cause PhD admissions committees to consider that candidate more favorably, assuming that he or she has the qualifications to be admitted?
I would guess that the answer is typically no. The administrators themselves certainly aren't looking through the list of Ph.D. applicants, and I doubt that they have any system for staff flagging such applications.
That said, they do schmooze a lot with donors (actually department chairs also schmooze more with donors than I realized), and if a donor had a child/relatively applying for a Ph.D. at the school, it would be totally natural for them to mention that to the chair of the department as an interesting fact. It would also be totally natural for the chair to mention that to the committee, though some probably wouldn't, precisely to avoid any appearance of pressure on the committee. I suspect that application would get a more careful look due to just curiosity. I doubt there would be any conscious attempt to tilt the scales, but simple attention might give them a small boost, but no more than if the parent was a friend of someone in the department, a well-known researcher in the field, or a famous person who hadn't made a donation. There are a lot of unfairnesses in the system based on how good the networks of your LOR writers are; I think that's a much bigger issue.
Since the admissions decisions are made by faculty, most of whom have tenure, rather than professional admissions staff, I think there's much less possibility for this sort of corruption; probably the indirect effect of donations getting students into prestigious undergrad institutions, and thus having an enhanced application is bigger.