1. A Kohain (priest) who has children with a divorcee or any other women who is forbidden creates challalim. They are not Kohanim (priests).
2. Just because my father’s name is Cohen, and my grandfather’s name is Cohen, does not stop me from legally changing my name to Smith.
3. Pinchas was not a Kohain (priest) until after the ma’aseh with Zimri. So until that point, he was not a Kohain (priest) even though his father (Elazar) and grandfather (Aharon) both were.