I think I was remiss in not answering Kismet's question "Do coaches have special privileges within admissions etc?"
Undergraduate admissions are are made at university level and coaches can and do have influence advocating for a student with the director of admissions. If a student athlete meets the minimum academic requirements they will usually get admitted. Worst case, there may be some some remedial course work required over the summer. Some undergrad programs are more selective, like programs in Whitman and Newhouse. So a student athlete may get admitted to SU, but may not initially get the academic program they want. But if their GPA is good enough, they can usually transfer to their desired program after their freshman year.
At the grad level, the coaches need to deal with the program directors (faculty responsible for grad admissions) for each grad program. There are dozens of grad programs so a coach has to deal with the director of the program that the student wants (or parent.) Yes, some coaches have influence with some program directors, but not others. The more selective grad programs have less wiggle room and the coach may not be able to move the needle and other less selective grad programs the coaches are welcomed with open arms; especially if the athletic dept is paying most or all of the tuition. And as you suspect, deals are made all the time, but rarely do we ever get a student that doesn't academically belong at the university.
So, yes coaches have a great deal of influence and they are reasonable and fun to work with.