No they do not..the genealogies of the Levitical line was kept in the Temple.. The Jewish priesthood, like the catholic priesthood ..was seen as a special set aside people as they had a special access to God that ordinary people did not have.. that "special "access was destroyed at the cross, the curtain was rent and access to God open to all that believe.. that is the priesthood of the believer..no more holier or set aside priests
Please read the book of Nehemiah to understand the absolute necessity for a PURE levitical line of Levites.
That is a false characterization of our teaching on the priesthood. The laywoman Catherine of Siena, and the "religious" sister Terese of Lisieux enjoyed a special intimacy with God, but one which is only proleptic. That is, all the blessed will enjoy such intimacy some day.
What priests have is a special authority.
This is a critical misunderstanding. It would indeed be outrageous to suggest that clergy, by virtue of their order enjoyed more blessedness than the laity. But, as has been pointed out before, not only did Dante put clerics and even popes in hell, but no one said it was heretical of him to do so.