> As a professor writing to another professor, should I use 〜さん (like in other workplaces) or 〜先生?

Stick to 先生. This is the same even if you become a university president or win a Nobel Prize.

> As a professor writing to a junior staff member (doctoral student or post-doc), what is the right form of address?

- If that staff member holds a title as a teacher (e.g., 教諭, 助教, 講師), always use 先生.
- If that staff member holds a license such as a medical license, which warrants being called 先生 even by the general public, then you use 先生, too.
- For post-doc researchers or research assistants who do not hold such titles or licenses, you can use さん in direct conversations or when addressing your subordinates in emails. Use 様 in emails to those outside your research group.

君 is still commonly used in some universities ([慶應義塾's convention][1] is well-known), but generally it's an outdated convention in academic fields.


  [1]: https://www.keio.ac.jp/ja/about/history/encyclopedia/18.html