If a Kohen or a Levite fathers a son with a gentile woman, does the son inherit the tribal status of the father despite not being halachically Jewish? If the son converts halachically, is he considered Yisrael?

Shalom, Thank you for your question. A child born to a non-Jewish woman is considered to completely follow the status of the mother, and does not inherit any of the status of the father. As such, in answer to your question, the child would be considered as a non-Jew before converting. If they converted, they would not have any of the status of their father – rather they would be a regular (non Cohen or Levi) Jew. Blessings.