Does every Jew who is a cohen or a levi use "Hacohen"/"Halevi" in his official Hebrew name (i.e. for a religious document)?
If one is the son or daughter of a cohen or a levi, does the "hacohen"/"halevi" enter one's name, too?
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The way the term is used is (Elazar ben Aharon) hacohen and not Elazar ben (Aharon hacohen). Thus the title applies to the person whose name is being stated and not to his father. As an example, if a baby (not Jewish) is adopted by a cohen or a Levi, then he is called to the torah as Gershom ben Moshe and not Gershom ben (Moshe haLevi).
An interesting point is that even though Moshe Rabbeinu was considered a Cohen, his sons were not cohanim but were leviim and would be called to the torah that way - (Gershom ben Moshe) haLevi.
A woman does not have this possibility, so she would be referred to in her kesuvah or her get as bas Ploni haCohen. As an example, my wife is a bas Cohen and is written that way in her kesuvah.
@SAH also points to the situation in which the child of a bas Cohen or bas Levi and the father is not Jewish. When such a child is called to the Torah with his grandfather's name, he is treated like an adopted child since he is not the member of a tribe.
This shows that the usage of the phrase is on the person and not the father. If the person's name is given without his father being mentioned then we do say Aharon haCohen
@DanF points out that misbeirach for a chole (sick person) is Aharon hacohen ben Yocheved