Yes, and so does the Bible, the Torah, and most other religions. The only difference between Muslims and most other followers of other religions, is that non-Muslims are much better at ignoring the "bad" parts of thier books.
Yes, but nearly all religions are like that. For instance, all of Leviticus is pretty ugly. The only reason I think that the Quran may be more violent than our present-day bible is because they have been using the original (or near original) book for the majority of their religion.
If you say that the Christian books don't encourage violence, then the Quran doesn't either.