Hajj on Someone's Behalf (Ahadith 1572 - 1574)

Bismillah.
Hadith no. 1570 and no. 1571 (below) are repeats. Read them here.
Volume 3, Book 29, Number 75:
Narrated Ibn 'Abbas (radiallaahu `anhu):
While a man was standing with the Prophet (sallallaahu `alayhi wasallam) at 'Arafat, he fell from his Mount and his neck was crushed by it. The Prophet (sallallaahu `alayhi wasallam) said, "Wash the deceased with water and Sidr and shroud him in two pieces of cloth, and neither perfume him nor cover his head, for Allah will resurrect him on the Day of Resurrection and he will be reciting Talbiya."

Volume 3, Book 29, Number 76:
Narrated Ibn 'Abbas (radiallaahu `anhu):
A man was in the company of the Prophet (sallallaahu `alayhi wasallam) and his she-camel crushed his neck while he was in a state of Ihram and he died Allah's Apostle (sallallaahu `alayhi wasallam) said, "Wash him with water and Sidr and shroud him in his two garments; neither perfume him nor cover his head, for he will be resurrected on the Day of Resurrection, reciting Talbiya."

Today's Ahadith:

Volume 3, Book 29, Number 77:
Narrated Ibn 'Abbas (radiallaahu `anhu):
A woman from the tribe of Juhaina came to the Prophet (sallallaahu `alayhi wasallam) and said, "My mother had vowed to perform Hajj but she died before performing it. May I perform Hajj on my mother's behalf?" The Prophet (sallallaahu `alayhi wasallam) replied, "Perform Hajj on her behalf. Had there been a debt on your mother, would you have paid it or not? So, pay Allah's debt as He has more right to be paid."

Volume 3, Book 29, Number 78:
Narrated Ibn 'Abbas (radiallaahu `anhu):
A woman from the tribe of Khath'am came in the year (of Hajjat-ul-wada' of the Prophet (sallallaahu `alayhi wasallam)) and said, "O Allah's Apostle (sallallaahu `alayhi wasallam)! My father has come under Allah's obligation of performing Hajj but he is a very old man and cannot sit properly on his Mount. Will the obligation be fulfilled if I perform Hajj on his behalf?" The Prophet (sallallaahu `alayhi wasallam) replied in the affirmative.

Volume 3, Book 29, Number 79:
Narrated 'Abdullah bin 'Abbas (radiallaahu `anhu):
Al-Fadl (radiallaahu `anhu) was riding behind the Prophet (sallallaahu `alayhi wasallam) and a woman from the tribe of Khath'am came up. Al-Fadl (radiallaahu `anhu) started looking at her and she looked at him. The Prophet (sallallaahu `alayhi wasallam) turned Al-Fadl's face to the other side. She said, "My father has come under Allah's obligation of performing Hajj but he is a very old man and cannot sit properly on his Mount. Shall I perform Hajj on his behalf? The Prophet (sallallaahu `alayhi wasallam) replied in the affirmative. That happened during Hajjat-ul-wada' of the Prophet (sallallaahu `alayhi wasallam).
It is permissible to do Hajj on behalf of someone else. It is also permissible for a woman to Hajj on behalf of a man, and vice versa. 
Ibn Qudaamah said in al-Mughni (5/27): 
It is permissible for a man to be appointed to do Hajj on behalf of a man or a woman, and a woman on behalf of a man or a woman, according to the majority of scholars.

How to perform Hajj on another's behalf?

Shaykh Ibn `Uthaymeen was asked:
If a person intends to do Hajj on behalf of someone who is unable to do it because of sickness, or who has passed away, how should he do the rituals? Should he choose Hajj Tamattu’ [where one does ihraam for ‘umrah then after ‘umrah comes out of ihraam and does a new ihraam for Hajj] or al-Ifraad [where one does Hajj only]?
He replied:
The one who is doing Hajj on behalf of another should say “Labbayka ‘an fulaan (Labbayka on behalf of So-and-so).” He should do Tamattu’ because this is the best of the rituals. Every person who is entrusted with something has to do what is best, unless the person who sent him has chosen something else, because he is a trustee and is obliged to do what is best.
Liqa’ al-Baab al-Maftooh by Ibn ‘Uthaymeen, 170
P.S. Comment on the narration about al-Fadl (radiallahu `anhu), where he looked at a woman and Prophet (sallallaahu `alayhi wasallam) turned his face away to avert his gaze, here.