Since I am not British, I can only give you an American point of view. I suppose what you have there would sound contemporary in Britain, but once again, I can only postulate.
Gate: I am not exactly sure what this entails in Britain. The closest thing I can think of is a student being put on "academic probation" which means that their grades are paid special attention to due to poor performance. Certain privileges may be revoked also.
Rusticate: Once again, I suppose this would be fine in Britain. However, I would think suspend is more contemporary and common.
Send him down: Expulsion is a much more common word. I have never heard "send down" as a saying, but it may be a popular euphemism in Britain.
Keep their noses to the grindstone: This seems to be a very sound idiom for the situation. It is a semi-common saying and simple enough that the meaning can also be inferred if it has not been heard before.
Student or Learner