Thanks 
To answer a few of your questions: Oxford/Cambridge are actually relatively transparent with their admissions processes. For Cambridge specifically, there is this page which contains a wealth of numbers about their process. For HSPS in Sidney Sussex it looks like there are usually 30-50 applicants with 3-7 acceptances each year.
I believe the feedback is given only to students who reach the interview stage. About a week after I got the rejection letter from UCAS in January, someone in Sidney Sussex’s admissions office blasted out a form that referees could fill out to receive feedback. I would assume that a sizeable number of referees do this (especially those actually in the UK and familiar with the process).
As an aside, I certainly hope that (elite) US unis will adopt a level of transparency similar to what the best UK unis offer. Although I can understand how opaqueness benefits admissions offices in furthering their goal of building diverse and capable classes for their respective institutions, I don’t think publishing statistics and providing comprehensive, individualized feedback has harmed Oxford and Cambridge - they’re still world-class institutions without a doubt, with classes that are plenty diverse and capable. On the other hand, the greater availability of information means there’s less frustration with “inexplicable”, “confusing”, or even “mistaken” decisions that seems to plague admissions to elite US unis. I think you’d be pretty hard-pressed to find frustration-, disillusionment- and disappointment-filled threads on a certain UK site similar to CC.
Hope that helps