I have seen the original question. BUT this part of the question has remained unanswered:
"if a Bas Kohain had a child with a non-Jew, later divorced him (or he passed away) then marries a Kohan and has a child with him; is this child a BECHOR?"
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It depends what relevance of the term "BECHOR" you are asking about:
If you mean BECHOR for Pidyon Haben, redemption of the firstborn son, the father does not have to redeem his son, even though the child is born from a forbidden union. (see Shulchan Aruch YD 305:19). Anyway there would be no pidyon haben because pidyon haben is only for the MOTHER'S firstborn, not the father's. (see ibid 305:17).
If you mean A BECHOR for double inheritance, He will receive his double inheritance as any Bechor would.