I think a lot of the question here is in the Administrator/Faculty divide. Is a special education teacher faculty? What about learning center, where students who need extra help in a specific subject go while the rest of the class continues regular learning? What about specialists who teach computer, science, garden, art, music? (And who give the teachers a break during the school day.)
As for administrators, the ones I see at schools are usually needed: a nurse or medical aide, a psychologist, a librarian, two front desk people (both so you have backup if someone is sick, and so you aren't without someone if a student/parent needs assistance out of the office), custodians are important (you need two, one for daytime, and one for nighttime. Yes, students can do cleanup at some point, but not K-3 at least), etc.
As for administrators, the ones I see at schools are usually needed: a nurse or medical aide, a psychologist, a librarian, two front desk people (both so you have backup if someone is sick, and so you aren't without someone if a student/parent needs assistance out of the office), custodians are important (you need two, one for daytime, and one for nighttime. Yes, students can do cleanup at some point, but not K-3 at least), etc.
Here's the staff directory of an elementary school near me. Which staff would you cut? http://www2.goleta.k12.ca.us/lapatera/staff-directory