1. Buddhists believe that death is a teacher. Would you agree?
Yes, to a certain degree. Death certainly has initiatic aspects -- which has been demonstrated in recent decades by studies of near death experiences. People returning from these experiences have a much broadened perspective and become more tolerant. They feel a greater connectedness with everyone and everything as a result of their experience. They usually have a changed perspective of death: from one of dread to a peaceful acceptance of death as a mere transition, another step on the path of evolution.
2. What are the advantages or benefits of meditating on one's own death? Are there disadvantages?
Among the benefits is becoming aware of one's own mortality and realizing how precious life is. It helps to keep things in perspective. The downside, of course, is the risk of becoming preoccupied with death, even to the point of becoming consumed with dread, which in turn hinders enjoyment of life. A balance must be struck between the two. The ideal which is displayed by many who've had near death experiences -- and which, I believe, would be the Buddhist ideal as well -- is neither a fear of death, nor desire for it; death is a transition.
3. Which do you think is the more important question: What happens when I die? or What is happening now?
Although contemplating what happens at the moment of death and thereafter can be a helpful exercise for some, it should not dominate our thinking. This is one of the faults of religions that focus too much on salvation and the afterlife while neglecting this life.