Good questions listed. I worked in admissions for private schools over 10 years. So, here are a few more. What is their rate of kids going to college (% that go on to 4 year schools) - what is their attrition rate from division to division? What is their college advising like? And here is a good one...how many hours of homework do they have per night? (the first school I worked at, the middle and upper school kids had 4 to 5 hours per NIGHT, plus required athletics. To my thinking, that leaves no time for being a kid. But yet, they sent more kids to Ivy League schools than any other school in our area). You could ask what schools the last 4 or 5 graduating classes went on to. What their one year college GPA average is.. that tells how well they learned study habits... Ask what other fees are required beyond tuition. (books, athletics, uniforms...lunch plan). Ask what percentage of teachers have advanced degrees. Ask how much of their budget goes to financial aid. Ask about re-enrollment (is it automatic from year to year once they are admitted, or do they have to pass or retest?)
Generally when you start putting some questions out there, the admissions staff will cover the whole area for you.