Go to the website of the respective institution and see what it calls itself.
Only Teacher's is wrong in your examples as more than one teacher is trained there.
The apostrophe can also be used for inanimate objects:
(Coffa, 6 Mar 2006)Please don't be fooled by over-prescriptive grammar books. This is a guide NOT a rule.
For example, it is perfectly acceptable to say "The car's headlights were out of alignment." In fact, the supposedly 'correct' form "The headlights of the car were out of alignment." is rare in actual usage.
Student or Learner