What was Sarah doing in Chevron, seeing as she and Avraham were living in Be'er Sheva?
Rashi (22:19): In fact, they resided in Chevron, and Avraham only went from the Akeidah to Be'er Sheva temporarily. 1
Ha'Emek Davar, Oznayim la'Torah, Malbim: Since Sarah was destined to die in Chevron
Why was Chevron called 'Kiryas Arba'?
Rashi #1: Due to the four giants who lived there, Achiman, Sheishai, Talmai and their father.
Rashi #2: Due to the four couples who were destined to be buried there - Adam and Chavah, Avraham and Sarah, Yitzchak and Rivkah, Ya'akov and Le'ah.
Oznayim la'Torah: Due to the four mothers who were buried there
From where did Avraham come?
Rashi: He came from Be'er Sheva. 1
Targum Yonasan: He came from Har ha'Mori'ah.
Where was Yitzchak? Why did he not participate in his mother's funeral and Hesped?
Targum Yonasan: Refer to 22:19:1:1*.
Why does the Torah juxtapose the death of Sarah and the Akeidah?
Rashi: Because, it was due to the news of the Akeidah, when she heard that her son was on the verge of being slaughtered, 1 that her soul left her and she died.
Why did the people of Ches delay burying Sarah until Avraham arrived?
Sanhedrin, 67b: To enable Avraham to eulogize her. This teaches us that eulogy is in honor of the deceased, and not of the living. 1
Torah Temimah (Ibid.): Otherwise, there would have been no justification in delaying the burial of Sarah for the honor of Avraham. See Torah Temimah's notes for clarification.
Why is the 'Kaf' in v'Livkosah" small?
Ba'al ha'Turim #1: Because, due to Sarah's advanced age, Avraham (Rosh - people) did not weep so much.
Hadar Zekenim, Riva: It hints that she lost 20 years (she should have lived to be 147, like Ya'akov 3 ) for asking Hashem to judge between her and Avraham.
Peninim mi'Shulchan ha'Gra: Avraham knew that she completed what she needed to in this world. He did not cry over her soul, only over her body. "Lo Tisgodedu... Al Mes Ki Am Kadosh Atah" (Devarim 14:1,2) (so you need not be so upset).
Ha'Emek Davar: It hints to little crying, i.e. less than the eulogizing. Refer to 23:2:151:3.
Ner Uziel (p.133): After Avraham passed the test of the Akeidah, the Satan attempted to get him to regret it, through causing that Sarah die through hearing about it. Avraham did not cry much, lest people say that he regrets the Akeidah.
Ba'al ha'Turim: And the Midrash in Eivel Rabasi explains that one does not eulogize someone who takes his own life (and similarly, we do not cry so much over them).
This is difficult, for all the Avos should have lived to be 180! Ya'akov lost 33 years for complaining to Paro. Moshav Zekenim (Bereishis 18:6) says that she should have lived to 175, like Avraham. I.e. had she not asked Hashem to judge, she would have deserved to live 180, and would have died five years earlier, lest she see Esav sin (PF).
Mo'ed Katan 27b says that three days are for crying, and [after, until day] seven, for eulogies. Why did Avraham first eulogize, and then cry?
Hadar Zekenim and Riva: He arrived more than three days 1 after she died. People of the household were already eulogizing, so he joined them, and cried later.
Ha'Emek Davar #1: When Avraham arrived many people were around the house. He eulogized immediately, so many would hear.
Ha'Emek Davar #2: "Li'Vchi ul'Misped" (Yeshayah 22:12) teaches that crying is first, when the death (or the Churban, which the verse discusses) changes the lifestyle of the survivors. Avraham's lifestyle did not change, for Yitzchak was already mature, therefore he eulogized first.
Malbim: Normally, first the mourners cry over the loss to the Niftar and his family, and this arouses eulogy. The eulogy over Sarah was for her deeds and perfection; this aroused crying!
We can resolve this with the Midrash that Sarah died from hearing from the Satan about the Akeidah. They walked three consecutive days to the Akeidah, and one of the next three days was Shabbos, so it took four days to return. (PF)