Fascinating. Aaron was also a Levite, if I'm not mistaken. But Aaron and his descendants were distinct among the Levites, and had greater authority.
We Mormons have two categories of "priesthood," Aaronic and Melchizedek. We believe that John the Baptist held the Aaronic priesthood, and thus had authority to baptize Jesus. I'm not sure whether that belief is unique to us, or not.
No, that is more-or-less the Jewish (and historical Christian) belief.
It was more of an issue when the Temple still stood, but the Kohain still do certain things things and face certain restrictions related to corpses and marriage.
As noted by BBL above, I would be dis-allowed as a Kohain because of my acceptance of Christ as the Messiah and am considered by several of my Orthodox family members to be a complete non-entity.