Student or Learner
Could we ever use "on" instead of "down" in these cases?:
- Tears ran down her cheeks.
-Tears ran down her face.
***Not a teacher***
In my opinion no. The use of running demands a direction. 'Tears ran down her cheeks/face' is a very standard construction. You could say 'she had tears on her face', but as soon as the tears are running, it really demands a direction.
If she was hanging upside down, you could probably even say 'The tears ran up/across her face'