Is there any effective difference between near perfect scores and perfect scores at schools that fit into the category mentioned above? (For example, 35 vs 36 or 790/780 vs 800)
If so, how large?
Since we don’t work on the admission boards for those schools we don’t know, but there seems to be some difference when you look at the Common Data Set. For Caltech http://finance.caltech.edu/documents/394-cds2015_final.pdf a 35 is at the 25%ile (and a 770 for SAT). It’s very possible that wasn’t a differentiator but we don’t know.
It seems to be less of an impact at MIT. http://web.mit.edu/ir/cds/2015/c.html
No. Variations that small are easily the result of a single oddly-worded question, or what the test-taker had for breakfast. As a rule of thumb, any SAT variation of less than 30 points is unlikely to make much difference.
If you want to demonstrate math ability, look at the AMC competitions. The difference between one point on the ACT is negligible.