I have been trying to find answer everywhere but no luck so far.
1) Does the effect of governor training will work only when he is appointed as governor?
2) Does the effect of general training will work only when he is appointed as general or sent on war?
3) Should I train all family members and court candidates with general and governor skills?
1. Yes
2. Yes
3. Depends on your short and long term planning. If you invest the money in those people, and you're planning to use them on those posts (for instance if you have 5 provinces total, you can use the king, the queen and 3 more great people). If you plan to have 1 more province in the near future, you can train one more. Again, it is a matter of investment and planning.