I was wondering what would be the status of a Ger with a Jewish father - would he now be ’related’ to his father in a Halachic sense, and would he have to take the Hebrew name ’Avraham Avinu’ or use his father’s name e.g. ’Avraham ben Yehudah’? Also, would he now be considered a member of his father’s tribe?
A child born to a non Jewish mother is non Jewish regardless to the father's status. Therefore all that applies to a convert applies in this case as well – halachacly he is not his father's son. He will be called "Ben Avraham Avinu" – at least for Gett and Ketubah matters. (Piskei Din Yerushalaim) The same applies for the tribe.