1) THE "SFEK SFEIKA" OF AN "ALMANAS ISAH"
QUESTION: Rebbi Yehudah ben Beseira rules that the daughter of an "Isah" is disqualified from marrying a Kohen, while the Almanah of an "Isah" is permitted to marry a Kohen. The Almanah is permitted to marry a Kohen because of the Sfek Sfeika about her status.
Why does the Sfek Sfeika not apply to the daughter of an "Isah" as well?
ANSWERS: The answer to this question depends on the two opinions in the Rishonim about the nature of the Sfek Sfeika in this case.
(a) RASHI explains that the Sfek Sfeika of the Gemara here differs from the normal concept of Sfek Sfeika. In this case, the husband has the status of an ordinary Safek Chalal (i.e. a Safek Ben Gerushah). The woman "acquired" this Safek when she married the Safek Chalal. Since her status of Safek is not due to a doubt about her own lineage, but rather she "inherited" this problem from her husband, the Gemara refers to her status as a Sfek Sfeika (in the sense that her status is a secondary source of Safek and not a primary source).
According to this explanation, the difference between the daughter of an "Isah" and the Almanah of an "Isah" is clear. The status of the man's offspring is not a secondary source of Safek but is an extension of the man's own Safek, since his offspring is an extension of him. Hence, the daughter has the same degree of Safek as her father. The Almanah, however, is not an extension of her husband, and she is more removed from the primary source of the Safek.
(b) TOSFOS in Kesuvos (14a) and the RAMBAM explain that the Sfek Sfeika refers to the case of a family of Kohanim in which there is a Safek Chalal, but the identity of that Safek Chalal is not known. When a woman marries any man from that family, she is faced with a Sfek Sfeika: perhaps the man she has married is the Safek Chalal and perhaps he is not the Safek Chalal, and even if he is the Safek Chalal perhaps he is really not a Chalal but a valid Kohen. This is the Sfek Sfeika in the case of the Gemara. (It is a normal case of Sfek Sfeika, in contrast to the explanation of Rashi.)
Tosfos in Kesuvos explains that the Sfek Sfeika indeed should apply to the daughter as well, but due to the principle of "Ma'aleh Asu b'Yuchsin" the Chachamim were more stringent with the daughter's status. The reason for this stringency is that the Almanah has a "Chezkas Kashrus" because she was permitted to marry a Kohen before she married this man. The daughter, in contrast, was born into a state of Safek and never had a Chazakah prior to the Safek. Although a Chazakah normally is not needed in order to rule leniently in the case of a Sfek Sfeika, the Chachamim were stringent with regard to the Safek of the daughter's status due to "Ma'aleh Asu b'Yuchsin."
The Acharonim question the explanation of Tosfos. The SEFER HA'MIKNAH asks that just as the concept of Sfek Sfeika applies to the woman, it should apply to the man whom she marries: perhaps he is the Safek Chalal in the family and perhaps he is not, and even if he is the Safek Chalal perhaps he is really not a Chalal but a valid Kohen. Hence, in a case in which a Safek Chalal became mixed into a family of Kohanim, the concept of Sfek Sfeika should determine that there is no Pesul in the family.
The answer to this question is that one of the laws of Sfek Sfeika states that both of the doubts must take effect simultaneously. If one Safek takes effect before the other, the principle of Sfek Sfeika does not apply and the case must be treated like a regular Safek. (See Shulchan Aruch YD 110.)
For the Almanah, both doubts occur simultaneously -- at the moment she marries the man. In contrast, for the Safek Chalal himself, the two doubts occur at different times. The doubt whether he is a Chalal or not occurred first. Afterwards, when the identity of the Safek Chalal became unknown, the new Safek -- who the Safek Chalal is -- occurred. This double doubt cannot be considered a valid Sfek Sfeika.
REBBI AKIVA EIGER (in Kesuvos) asks another question on the explanation of Tosfos. Why does the Chazakah of the mother not validate the status of her daughter? Tosfos earlier in Kidushin (66a, DH Mai) discusses the concept of "Chezkas ha'Em Mehani l'Bas" -- the Chazakah that exists for the mother applies to her offspring. Why does the Gemara here differentiate between the Almanah and her daughter on the basis of the lack of a Chazakah? On the contrary, the same Chazakah should apply to both women.
Rebbi Akiva Eiger answers this question as follows. In this case, the daughter has two opposing Chazakos from her two parents. Although she has a Chezkas Kashrus from her mother, her father is a Safek Chalal. Since there are two opposing Chazakos to follow, one cannot be chosen over the other. (A. Kronengold.)

75b----------------------------------------75b

2) A MARRIAGE OF KUSIM
QUESTION: The Gemara (75a) cites a Beraisa in which Rebbi Elazar states that a Kusi may not marry a Kusis. The Gemara here explains that Rebbi Elazar follows the view of Rebbi Yishmael that Kusim are "Geiri Arayos" who converted with impure motives and thus are considered like Nochrim. Since Rebbi Yishmael rules like Rebbi Akiva that the child of a Nochri and Jewess is a Mamzer, Rebbi Elazar states that a Kusi may not marry a Kusis because one of them might be a Mamzer.
RASHI explains that the doubt is that one of the two Kusim may have been born to a Kusi who married a Jewess and is thus a Mamzer, while the other may have come from a pure family of Kusim who married Kusim and is not a Mamzer but a Nochri, and a Jewish Mamzer is prohibited from marrying a Nochri.
Rashi's explanation implies that the problem of a Kusi who marries another Kusi is that one Kusi may be a Jewish Mamzer (born to a Nochri who married a Bas Yisrael) who now is marrying a full-fledged Nochri. If that is the problem, however, why does the Gemara need to say that he might be a Mamzer? Even if the offspring of a Nochri and a Bas Yisrael is a legitimate Jew he is not permitted to marry a Nochri! Why does the Gemara need to say that Rebbi Yishmael rules like Rebbi Akiva who states that the child of a union between a Nochri and a Jewess is a Mamzer? Even if he is a full-fledged Jew the same concern exists.
ANSWERS:
(a) The RITVA quotes "Yesh she'Tirtsu" who explain that the Gemara is not discussing the first generation of Kusim who were not genuine converts. Rather, it is discussing subsequent generations of Kusim who are genuine converts. If both parents of the Kusi were Nochrim (since their conversion was not sincere) but he converts properly, he becomes a fully valid convert. If, however, the Kusi comes from a union between a Nochri (an insincere Kusi) and a Bas Yisrael, according to Rebbi Akiva the Kusi is a Mamzer and may not marry another Kusi (who has the status of a proper convert).
The Beraisa earlier (74b) cites a dispute between Rebbi Yosi and Rebbi Yehudah regarding whether a convert may marry a Mamzer or not. The Gemara here follows the view of Rebbi Yehudah who prohibits a convert from marrying a Mamzer. Hence, the Kusi who is a proper convert may not marry another Kusi because the other Kusi might be a Jewish Mamzer. (See MAHARSHA to Tosfos, 75a, DH Kusi, and PNEI YEHOSHUA to Rashi here.)
(b) TOSFOS cites the RI who offers an alternative explanation for the Gemara. Just as there is concern that Jewish women married into and became mixed with the Kusim, there is concern that Jewish men became mixed with them. Consequently, there might be a full-fledged Jew among the Kusim, and if he marries another Kusi he will be marrying a Kusi who is a Mamzer. Everyone agrees that a Jew may not marry a Mamzer.