Your summary of the event is good, except two things I'd like to comment on, Insha-Allah.
(1) As for referring to Ali [ra] as "brother", then the Prophet [s] similarly addressed Abu Bakr [ra] as his brother, and this is recorded in the Sahihayn. I can pull out the quote if you wish, Insha-Allah. The Shia arguments are funny because they will find a quote where the Prophet [s] calls Ali [ra] to be his brother, and no Sunni will deny this. And yet, since when can you only have only one brother???? And even more than this, if the Shia would like to claim Ali's right over Abu Bakr simply because the Prophet [s] called Ali [ra] his "brother", then Abu Bakr [ra] has an even greater right since the Prophet [s] called Abu Bakr [ra] not only his brother but also his father!! And we all know that "father" is a higher position than brother!The third position was held by those that supported Ali ibn Abu Talib, as he was his relative, and to whom the Prophet adressed several times with "my brother" and "my heir".
(2) As for the claim that the Prophet [s] called Ali [ra] as "his heir", this is no doubt based on weak and inauthentic narrations! So we do not accept this!
Yes, but more specifically, Ali [ra] was doing Ghusl of the Prophet's body, a task which is reserved for a handful of close relatives. The rules in Fiqh say that as FEW people as possible should be in the room when the body is washed, in order to maintain the Haya of the body. And Fiqh further specifies that the close relatives should do it. We read:Ali, on this time, being a member of the Prophet's family was busy preparing for the funeral.
“The dead body should preferably be washed by a relative.”
(Everyday Fiqh, Vol.1, by Abdul Aziz Kamal, http://muslim-canada.org/fiqhch29.html)
“At the washing, only people whose presence is needed may stay.”
(Fiqh-us-Sunnah, Vol.4, Sayyid Saabiq)
And we read further:
"The majority of jurists are of the opinion that washing the body of a dead Muslim is a fard kifayah or a collective obligation. If some people attend to it, it is done on behalf of all, as commanded by Allah’s Messenger, peace be upon him, and practiced by the Muslim community."
(Fiqh-us-Sunnah, Vol.4, Sayyid Saabiq)
In other words:
1. The Prophet’s body should have been washed by his close relatives.
2. No other extra person should be present except those absolutely necessary from amongst his relatives.
3. The obligation to wash the Prophet’s body is a communal obligation; the Prophet’s relatives removed any obligation from the shoulders of the rest of the community, including Abu Bakr (رضّى الله عنه) and Umar (رضّى الله عنه) .
My point is that Abu Bakr [ra] ONLY missed the Ghusl, not the FUNERAL, which he most definitely attended. All of this means that the Shia are barking for no reason. Abu Bakr [ra] only missed the Ghusl, something he [ra] couldn't take part in any way. As for the funeral, Abu Bakr [ra] was most definitely there for that.
Not only did Abu Bakr (رضّى الله عنه) help out with the burial, he was actually the one who is credited with deciding where the Prophet (صلّى الله عليه وآله وسلّم) was to be buried. We read:
" The task of washing the body being over, the Companions were divided over the place of burial. Abu Bakr then said: “I have heard from the Messenger of Allah that every Prophet is buried at the spot where he has breathed his last.” The Prophet’s bedding was accordingly removed from the place and a grave was dug for him at the spot. "
(Tareekh al-Islam, Vol.1, p.246)
I don't think Abu Sufyan [ra] asked Ali [ra] to take Baya'ah to Abu Bakr [ra], but rather Abu Sufyan [ra] said that he himself was ready to swear allegiance to Ali [ra] and back him with soldiers AGAINST Abu Bakr [ra].But later on, when Abu Sufyan approached him and summoned to take the oath of allegiance to Abu Bakr,
I think maybe you just had a typo there. So just clearing it up.
Anyways, otherwise your summary was good. I don't mean to nitpick. Just wanted to make it a bit clearer Insha-Allah.