A child who is born from the union of a Ger and a Manzeres is a Mamzer.
It is a Machlokes Tana’im if a Ger is permitted to marry a Mamzeres.
If a Chalal marries a Bas Yisrael it is a Machlokes if the daughters born from that union are Pesulah for Kehunah. (1)
If a Yisra’El marries a Chalalah the daughters the daughters born from that union are Kosher for Kehunah
R. Yochanan holds that if a Mitzri Sheni (second generation Ger) marries a Mitzris Rishonah (first generation Ger) the child is a Mitzri Shlishi and may marry into Klal Yisrael. (4)
R. Dimi argues with R. Yochanan and he holds that if a Mitzri Sheni marries a Mitzris Rishonah the child is a Mitzri Sheni because the status of the child goes after the mother, not the father.
If a man who is not from the seven nations has a child with a woman from one of the seven nations it is permitted to buy that child as a servant. (3)
If a man from the seven nations has a child with a woman who is not from the seven nations it is forbidden to buy the child as a servant.
A BIT MORE
1. A Chalal is someone who is born from Pesulei Kehunah (such as a widow who married a Kohen Gadol) and he is not valid for Kehunah.
2. A Ger Mitzri may not marry into Klal Yisrael; only the third generation may marry into Klal Yisrael. When a first generation Mitzri marries a second generation Mitzris R. Yochanan holds that since the Kidushin is permitted the status of the children follows the father, not the mother and since the father is a Sheni the child is a Shlishi
3. The Pasuk that states ‘do not allow any Neshamah to live’ which applies to the seven nations does not apply if only the mother is from the seven nations.