All of the above. And some schools have more structure, such as assigning supervised study halls during free periods, whereas others let kids decide how to use free periods. Some kids might chafe at the former whereas some kids might need it.
Some schools are very competitive whereas others are far less so. Some may be more artsy or more sporty. Some may have a culture in which wealth is normalized. These things are neither good nor bad but your kid might fit better with some of those.
As for the coach, it sounds like your inquiry should be around how your child would best be served in terms of sport. Looking to stay with same out of school elite team? Will this be in lieu of school team or does school require participation on their team? What’s the policy on missing class for non school sponsored sports? I think you can, and should, understand this if it pertains to you and it may lead you to the recruiting question.