Are private schools required to contact you after they receive your application to set up interviews and school visits?

They should, but if you sent the application in and they haven’t contacted you in a reasonable amount of time, call them.  You can say that you are calling to confirm that they received the application.  As long as you have them on the line, go ahead and set up the appointments.

In some cases (New York City, for example), most private schools use an online program such as the Ravenna Hub, through which you can apply and set up all of your own appointments.