There are at least some day cares and preschools that are staying open. I think it's a horrid idea, but the people running them just care about profits, not the health of society. 

For the love of mankind, at least put out a strong letter telling parents to not send their kids there if their jobs aren't 100% essential to society functioning (Dr, police, nurse, etc). If you work for an insurance company and can work from home, stay the fuck home and keep your kids there too.

I don't think private colleges are open either. Wake Forest University is closed at least.

Since all public colleges and K-12 schools are closed, are there any private colleges or K-12 schools still having in-school classes?

private colleges and schools