1. He begins to dial all his friends, starting with the names at the top.
  2. My mother, and my teachers starting with kindergarten, told me to do exercise.
  3. We had our examinations for the year 2010 starting from May 13th.
    Please check all the sentences and comment.

I would say “starting on May 13th”. Depending on context, it may be easier to say “Our examinations for the year 2010 started on May 13th”.

Otherwise, what you have written looks OK.