Gemara

78b: Sha'ul killed the Kohanim of Nov (who used to supply the Giv'onim with food and water). David went to appease the Giv'onim. They demanded "... give us seven of Sha'ul's children, and we will hang them."

79a - Question: "From the beginning of the harvest ... until the rain (the bodies were hanging)". It says "You may not let him (someone executed) hang overnight"!

Answer (R. Yochanan): It is worth uprooting a letter of the Torah in order to publicly sanctify Hash-m's name;

Passersby would ask about the people hanging. They would learn that they were sons of kings who harmed converts.

Passersby: There is no nation as fit to cling to as Yisrael! If sons of kings are punished thusly, all the more so, commoners! If for harming converts thusly, for harming Yisrael all the more so!

Immediately, there were 150,000 new converts - "Shlomo had 70,000 workers carrying on the shoulder, and 80,000 quarrying in the mountain.

24a (Mishnah): If one was suspected of relations with a slave or Nochris and she converted, he should not marry her. If he did, he may stay married.

(Gemara) Inference: If she converts, the conversion is valid.

Contradiction (Beraisa - R. Nechemyah): If one converted to marry a woman or man, or to receive food or authority from the king, the conversion is invalid;

If one converted to be saved from lions, because of a dream, or in the days of Mordechai and Esther (when people feared the Yisraelim), the conversion is invalid, unless he converts nowadays.

Objection: This cannot be!

Contradiction: Rather, it must be like nowadays (when it is surely not for authority).

Answer: R. Yitzchak bar Shmuel said that the Halachah follows the opinion that all these are valid conversions.

Question: If so, he should be allowed to marry her!

Answer: One should not cause people to speak ill about him;

(Rav Asi): (The Mishnah forbids marrying her due to) "Remove from yourself stiff talk and ill rumors".

(Beraisa): We do not accept converts in the Messianic period. Similarly, we did not accept in the days of David and Shlomo.

(R. Elazar): We learn from "Hen Gur Yagur..." (only one who dwells with us (converts) when Hash-m is not with us, i.e. in our poverty, will receive a portion with us in the world to come).

47b (Beraisa): After immersing, he is a full Yisrael.

This teaches that if he reverts (to serve idolatry) then engages a Bas Yisrael, he is considered a heretic Yisrael, and his Kidushin takes effect.

Rishonim

Rif and Rosh (16b and 4:36): After immersing, he is a full Yisrael. This teaches that if he reverts (to idolatry) then engages a Bas Yisrael, he is considered a heretic Yisrael, and his Kidushin takes effect.

Nimukei Yosef (DH Kidushav): Presumably, he is like a Yisrael regarding all monetary laws. However, if the kingdom says that he is like a Nochri, the law of the kingdom is binding.

Rosh (2:6) If one was suspected of relations with a slave or Nochris and she converted, he should not marry her. If he did, he may stay married. The conversion is valid even though it was to marry a man. The Halachah is, conversion in order to marry a man or woman or to receive food or authority from the king is valid.

Rambam (Hilchos Isurei Bi'ah 13:14): Do not think that Shimshon, who saved Yisrael, or Shlomo ha'Melech, who was called Yedidyah (Hash-m's beloved) married Nochriyos. Rather, it is proper that when a Nochri comes to convert, we check if his intent is to obtain money or authority, or due to fear of something or in order to marry a Yisrael(is). (If it is, we do not accept him.)

Rambam (15): This is why Beis Din did not accept converts all the days of David and Shlomo. In the days of David, perhaps they converted due to fear; in the days of Shlomo, perhaps it was due to the kingdom and the great benefits Yisrael had at the time. Anyone who converts due to one of the follies of the world is not a Ger Tzedek. Nevertheless many converted in front of commoners in the days of David and Shlomo. Beis Din ha'Gadol was suspicious about them. They did not reject them, since they had done Tevilah, but still they did not draw them close until observing them.

Magid Mishneh: There were 150,000 converts after people saw how the Giv'onim were appeased. The Rambam explains that they converted in front of Batei Din of commoners.

Beis Yosef (YD Sof 268 DH v'Al): The Rambam discusses conversion in front of commoners, not a Beis Din of commoners. Nowadays we are considered commoners (Gitin 88b). L'Chatchilah we do not accept insincere converts, but b'Di'eved it is valid. This is like the Mordechai.

Hagahos (brought in Frankel Rambam): Some defend the Magid Mishneh. The Rambam's words 'Beis Din ha'Gadol was suspicious' implies that they were converted by Batei Din of commoners. Others reject this.

Rambam (16): Shimshon and Shlomo converted the women before marrying them. It was known that they converted for other reasons, and it was not according to Beis Din, so the verses consider it as if they married forbidden Nochriyos. Also, in the end the women served idolatry, proving that their conversion was insincere.

Rambam (17): If we did not check a convert's motivation or inform him of Mitzvos and their punishment and he converted in front of three commoners, he is a convert, even if it is known that he has an improper motive. After Tevilah he is not a Nochri; we are concerned about him until we verify his righteousness. Even if he reverts to serve idolatry, he is like a Yisrael apostate. If he is Mekadesh a Yisraelis it takes effect. This is why Shimshon and Shlomo kept their wives after realizing their insincerity.

Tosfos (24b DH Halachah): Here we conclude that conversion due to lions is valid. This is unlike the Kusim. They never abandoned their idolatry (this shows that their conversion was totally insincere). The opinion that they are true converts holds that they later converted sincerely.

Tosfos (24b DH Lo): The 150,000 converts in the days of David converted by themselves. Even in the days of David and Shlomo we took converts if it was clear that they do not seek benefits, e.g. Iti ha'Giti and Bas Paro. Hillel converted a Nochri who aspired to be Kohen Gadol, because Hillel knew that he will eventually serve l'Shem Shamayim. Similarly, Rebbi converted a (former) harlot who wanted to marry a certain Talmid.

Poskim

Shulchan Aruch (YD 268:12): When a Nochri comes to convert, we check if his intent is to obtain money or authority, or because he fears something, or he/she seeks to marry a Yisraelis/Yisrael. (If so, we do not accept him.)

Beis Yosef (DH v'Da): Beis Din may accept a convert with an ulterior motive if it assesses that he will eventually serve l'Shem Shamayim.

Shulchan Aruch (ibid.): If we did not check a convert's motivation or inform him of the reward and punishment of Mitzvos and he converted in front of three commoners, he is a convert, even if it is known that he has an improper motive. After Tevilah he is not a Nochri; we are concerned about him until we verify his righteousness. Even if he reverts to serve idolatry, he is like a Yisrael apostate. If he is Mekadesh a Yisraelis it takes effect.

Beis Yosef (DH Kosav): BaHaG says that if a convert reverts to serve idolatry, his wine and bread are forbidden like wine and bread of Nochrim. Presumably he learns from Yevamos 45b, which says that if a convert reverted to his old way he is a Min (priest of idolatry).

Prishah (35): This is obvious! The same applies to any Yisrael that serves idolatry! Some say that BaHaG forbids his wine retroactively, even before he reverted to idolatry. Alternatively, perhaps we forbid his wine even if it is Mevushal.

Rejection (Taz 16): Once he converted he was Muchzak to be a Yisrael. We cannot disqualify him retroactively. BaHaG teaches that the stringencies of Nochrim apply to him, but not leniencies, e.g. his Kidushin takes effect.

Teshuvas Radvaz (4:289 DH Teshuvah): Nowadays, all the Shifchos do not sincerely accept the Mitzvos. They merely say that they do to avoid being sold to Yishmaelim. We see that when things become difficult they adopt the rites of the Yishmaelim! Nevertheless, they are converts (i.e. Shifchos Kena'aniyos). Even converts due to lions are proper (Kena'ani) slaves; the Isurim of Yayin and Bishul Akum (food cooked by Nochrim) do not apply to them. However, we see that they do not keep the Mitzvos of women. I say that one should not rely on them for anything that requires exertion.