Gemara

(R. Chiya bar Ami - Beraisa): If a wife died during Kidushin, her husband is not considered an Onen (a mourner before burial of the Mes) and he need not become Tamei amidst burying her;

Similarly, if he died she is not an Onenes and need not become Teme'ah.

Mo'ed Katan 20b (Beraisa): One is Mis'avel (conducts Aveilus) upon losing any relative for whom the Torah commands a Kohen to become Tamei, i.e. a wife, parent, (paternal) brother or (virgin) sister, or child;

Chachamim added a brother or virgin sister from the mother, and a married sister from either parent.

R. Akiva says, he is also Mis'avel for the next generation (a grandfather, grandson or nephew);

R. Shimon ben Elazar says, (in the next generation) he is Mis'avel only for his father's father and his son's son;

Chachamim say, if Reuven is Mis'avel (mid'Oraisa) for Shimon, Reuven is Mis'avel with Shimon (whenever Shimon is Mis'avel mid'Oraisa).

Question: Chachamim agree with R. Akiva!

Answer: Chachamim require Reuven to be Mis'avel only with (in the presence of) Shimon (and R. Akiva does not distinguish);

Rav and Rav Huna told their sons to be Mis'avel (when their wives are in Aveilus) only in the presence of their wives.

Mar Ukva thought that he would observe Shivah and Sheloshim when his wife's brother died.

(Rav Huna): One is Mis'avel for his wife's honor only when she loses a parent:

(Beraisa #1): If a woman lost a parent, her husband cannot force her to color her eyes and be Pokeses (apply rouge to her face; alternatively, part her hair). Rather, he inverts his bed and mourns with her;

Similarly, if a man lost a parent, his wife may not color her eyes or be Pokeses. Rather, she inverts her bed and mourns with him.

Contradiction (Beraisa #2): Even though a man cannot force his wife to color her eyes and be Pokeses, she mixes his drink, makes his bed and washes his hands, feet and face. (Beraisa #1 says that that he inverts his bed even during her Aveilus!)

Resolution: Beraisa #2 discusses when she lost a relative other than a parent. Then, he is not Mis'avel with her.

Support (for Rav Huna - Beraisa): One is Mis'avel for his wife's honor only when she loses a parent:

Rishonim

Rif and Rosh (Mo'ed Katan 20b and 3:35): If Reuven is Mis'avel for Shimon, Reuven is Mis'avel with Shimon in Shimon's house. Rav and Rav Huna told their sons to be Mis'avel (with their wives) only in the presence of their wives. This is only when she loses a parent, but not for a sibling. Similarly, if he lost a parent she mourns with him.

Nimukei Yosef (DH v'Achoso): One is Mis'avel with his wife to show that he is pained by her sorrow. Therefore, it is only for her parents. One is not Mis'avel with a relative mourning a spouse. Some say that one is not Mis'avel (with someone) over someone he is not related to, e.g. the maternal brother of his paternal brother.

Nimukei Yosef (DH v'Achoso): If one of the seven relatives for whom Aveilus is mid'Oraisa (a parent, child, sibling or spouse) died on Yom Tov Sheni, Aveilus overrides Yom Tov Sheni, which is mid'Rabanan. One would not mourn on Yom Tov Sheni for relatives that Chachamim added, and all the more so one does not mourn with an Avel, for this is mid'Rabanan. Aveilus mid'Rabanan of an individual does not override Yom Tov mid'Rabanan of the Rabim.

Rosh: Nowadays we are lenient and do not Mis'avel with Avelim. The Gemara connotes that we are Mis'avel with an Avel for the honor of the Avel, and if he pardons his honor it is not necessary. Nowadays all Avelim pardon their honor.

Rambam (4): If Reuven would be Mis'avel over Shimon, mid'Rabanan Reuven is Mis'avel with Shimon in front of Shimon. E.g. if the son, brother or mother of Reuven' son died, Reuven must tear and observe Aveilus in front of his son. He is exempt when not in front of him. The same applies to other relatives.

Rambam (5): Even though one is Mis'avel over his wife after Nisu'in, he is Mis'avel with her only if she lost a parent, for her honor. Similarly, she is Mis'avel with him when he lost a parent, but not for other relatives.

Poskim

Shulchan Aruch (YD 374:6): If Reuven would be Mis'avel over Shimon, Reuven is Mis'avel with Shimon if a relative of Shimon died. This is only in front of Shimon. Not in front of him he need not be Mis'avel.

Bedek ha'Bayis (DH v'Chosav): If Shimon is Mis'avel with his brother Reuven, Shimon children need not be Mis'avel with their father, since Shimon is not Mis'avel over his own loss.

Shulchan Aruch (ibid.): However, even though one is Mis'avel over his wife, he is Mis'avel with her only if she lost a parent, due to the honor of his father-in-law or mother-in-law. Similarly, she is Mis'avel with him when he lost a parent, but not for other relatives.

Beis Yosef (DH v'Chosav Od): Some say that the Beraisa equates her Aveilus with him to his Aveilus with her, i.e. it is only for parents. Others say that since the Beraisa says 'for the honor of his wife is only for parents' but does not say 'for the honor of her husband is only for parents', that she is Mis'avel with him whenever he is an Avel. The Beraisa discusses when he lost a parent to parallel the case of his Aveilus with her.

Rema: Some say that nowadays we are lenient about being Mis'avel with Avelim. It is only for the honor of the Avel, and all are accustomed to pardon their honor. Nowadays the custom is not to Mis'avel with Avelim at all. One who is stringent is appalling.

Shach (5): Even though the Rema (at the end of this Halachah) permits one who is exempt to be Mis'avel, this is only if he totally observes Avelus. If he conducts Aveilus only in front of the Avel, this is like mocking him.

Rema (ibid.): Nevertheless, the custom is that all relatives of the Mes who are disqualified to testify about him show some signs of Aveilus the entire first week, i.e. until after Shabbos. They do not bathe before Shabbos, and leave on some weekday garments, unlike a regular Shabbos. Some places are stringent about different matters. What I wrote is primary. This refers to a Shemu'ah Kerovah, or one who was at the death. There is no source for this Aveilus over a Shemu'ah Rechokah. If one wants to be stringent and be Mis'avel when he is exempt, or to wear black over a relative, we do not protest.

Gra (8): Letter of the law, they should observe all laws of Shivah and Sheloshim. We find that Mar Ukva intended to do so.

Shach (7): A relative who (according to the Gemara) should be Mis'avel with the Avelim should be stringent not to bathe in hot water or eat outside his house until Motza'ei Shabbos. He should wear weekday clothes on Shabbos, except for a white cloak to honor Shabbos. Those who need not be Mis'avel with the Avelim are less stringent. In any case we follow the local custom.