I used the same open-door policy you mention for one-on-one meetings with students when I was a young university prof in 1992 – I think that’s adequately public and transparent for dealing with young adults at college or university, and is probably suitable for high school students as well (as long as it’s during regular school hours when people are likely to be passing in the hall).

Email at college is a different situation than high school — college students are adults (if still young and vulnerable ones), and they have a right to privacy even from their parents. Most high schools deal with email risk by requiring teachers to use Board-supplied accounts, which, presumably, can be monitored and audited (I doubt that happens, except after an accusation); personally, I would also CC the parents on any correspondence, unless there were some concern about the parents themselves that prevented that, in which case I would CC the principal or guidance councilor.

My daughter routinely notifies her college instructors by email when she must miss a class and expects to get a reply by email telling her what the homework assignment was. This was also true in high school (though less so — fewer teachers disclosed their email addresses then), so it’s not just a matter of her age.