Joseph, I don’t remember the exact details; his father was a Ba’al Teshuva, and I think his mother was a Giyores. Presumably, his father subsequently found out that he was a Kohen.
Laskern, yes, they would be. I don’t believe there is any question about this. (There was a debate among the Poskim about fifty years ago regarding the status a child from artificial insemination using donor sperm. Rav Moshe Feinstein ruled that he is not a Mamzer because the Mamzeirus status depends on a מעשה ביאה).