First question: when does a ger celebrating his Jewish birthday – on the Jewish date of his birth or on the Jewish date of his giyur?
Second question: a ger with Jewish father – is this ger’s Jewish name “ben his father’s Jewish name” or “ben Avraham Avinu”?
1. He can celebrate both. One day to thank H-shem for giving him his neshoma, and this is surely the main birthday, however he can also celebrate on the day that H-shem gave him his physical body.
2. His name is Ben Avrohom Avinu, and not his physical father’s name, since when he became a ger he is considered reborn.
Tzitz Eliezer 19-45, Tevous Shemesh 169-26.