The concept of Sharia, by the way, sounds a lot like Halakha in Judaism, and what Ahmed refers to as mystical Islam sounds a lot like Reform Judaism.
Wonderment, your first point is correct in that Shari'ah is the equivalent term in Islam to halakha in Judaism.
As to the second point, there is a certain amount of validity, but the problem as Ahmed actually alludes to in saying that violence could come from any of his three categories, is that one's approach to theology and one's approach to politics is not always the same. Which is to say, "mystical" Islam or Sufism can be more analogous to Hasidism, in that a mystical understanding of theology or the nature of God, is not at all necessarily indicative of a liberal or progressive approach to religious practice. There are Sufi Muslims who follow Shari'ah closely and Sufi Muslims who do not. Just as historically there have been Sufi Muslims who were pacifist and Sufi Muslims who were well known leaders of Jihad (the military kind).