When does an onen make Havdalah?

For the most part, an onen is obligated in mitzvos on Shabbos and Yom Tov. The poskim disagree about whether learning Torah is permitted.  If an issue arises regarding the funeral or burial on Shabbos or Yom Tov, the onen is exempt during the time that he needs to take care of the matter. It is permissible to wait by the techum border on Shabbos in order to tend to the needs of the deceased immediately after Shabbos. During that time at the border, the onen is exempt. When a person is not busy with the burial, the poskim disagree about whether he should say Kriyas Shema and daven Ma’ariv before the conclusion of Shabbos. Mishnah Berurah rules that one should say Kriyas Shema without the brochos. Havdalah is said after the burial takes place; it can be recited any time before Tuesday evening.

סעיף ב וס"ק ט ו־יא; ביאורים ומוספים דרשו, 23 ו־2

Can an onen daven in shul on Yom Tov morning?

On the first day of Yom Tov, it is permissible to ask a non-Jew to bury the deceased. On the second day of Yom Tov, the burial may be done by Jews. Either way, the relatives are still in the category of onenim and are exempt from mitzvos. On a weekday, relatives are in the category of onenim until the burial, even if an issue arises that holds up the burial. On Yom Tov, however, relatives are only exempt when there are practical things to be done to take care of the burial. Therefore, the poskim say that when everyone is in shul (and the Chevra Kadisha is not available) at night and in the morning, the relatives are obligated in mitzvos.

ביאורים ומוספים דרשו, 29 ,30 ו־3

Which tefilos may be said next to a mes?

It is forbidden to daven, learn, or say a brocha within four amos of a dead body or grave. Doing mitzvos is a form of mockery to the deceased, who can no longer perform mitzvos. According to some Rishonim, if one did daven, he is not yotzei and must daven again. Other Rishonim disagree. L’halacha, Kriyas Shema must be repeated, but not brochos or Shemoneh Esrei. Kaddish and other tefilos recited on behalf of the deceased may be said in his proximity. Some poskim do not permit the recitation of Tehillim, even for the protection of the departed.

סעיף ז וס"ק טו-יז; ביאורים ומוספים דרשו 29,30 

• An onen--a person who lost one of the seven close relatives and has not yet buried the deceased--may not perform any mitzvos. The Talmud Yerushalmi explains that he may not be involved in mitzvos because he must be involved in the burial arrangements. Also, doing other mitzvos gives the impression that a person is indifferent about the passing of a close family member.

• Although an onen is exempt from mitzvos, he may not engage in forbidden activities. He does not make brochos over food, since it is only forbidden to eat without a brocha if there is a mitzvah to recite a brocha.

• After the first layer of dirt is poured over the body, the mourner becomes obligated in mitzvos. If it is close to zman Kriyas Shema or tefilah, he must daven immediately, so as not to forfeit these mitzvos. 

• Are levayah attendees obligated to say Kriyas Shema?

• Can a burial be pushed off until after sunset?

• How long must one accompany a levayah?