Yes. To enroll in a boarding school, an application must be filled out and approved. From there, tuition must be paid, thus making all boarding schools considered to be private.
Absolutely not! There are a ton of private schools that have no affiliation with the church at all! Usually from Pre K-5th grade, they are only religious, but Prep schools (6/7-12) can be either way.
"Hillsdale Academy is not a boarding school, however it is a private school. Hillsdale Academy offers elementary and secondary education and is co-ed."