All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad, sallallaahu 'alayhi wa sallam, is His slave and Messenger.
First of all, we ask Allah, The Almighty, to reward you generously for your keenness to seek knowledge and ask about what confuses you regarding the matters of your religion; this is a unique quality of a believer. May Allah increase your keenness.
There is an old scholarly difference of opinion among Muslim jurists regarding this issue, and their discussions about its evidence go way back. The Hadeeth of ‘Aa’ishah that you mentioned was among the proofs brought forth by the majority of the scholars to support their view regarding stipulating the Wali (guardian) as a condition for the validity of the marriage contract. This is supported by many verses of the Quran, including the verse wherein Allah, The Exalted, Says (what means): {…do not prevent them from remarrying their [former] husbands…} [Quran 2: 232]
The verse addresses men regarding giving women in marriage. If a woman was entitled to give herself in marriage, then prevention from marriage by her Wali (mentioned in the verse) would not apply.
As for the report narrated on the authority of ‘Aa’ishah which Imam Maalik, cited in Al-Muwatta’, the majority of the scholars interpreted it to mean what you mentioned; that she only made the arrangements for the marriage as far as the engagement, bride's gift, and the like are concerned, because she said: “Conduct the (marriage) contract; for women do not conduct marriages.” This means that she did not conduct the marriage contract herself, nor did any woman.
It is not possible to say with certainty that her brother ‘Abdur-Rahmaan did not delegate her to act on his behalf in this regard. In his explanation of Al-Muwatta’, Al-Baaji mentioned the possibility that her brother ‘Abdur-Rahmaan had delegated someone to act on his behalf and the agent conducted the marriage contract with the bride's gift less than what was believed to be given to (a woman under the guardianship of) a man of his standing.
Al-Baaji suggested the possibility that the one who conducted the marriage contract was his brother or son because he was in charge of his affairs (in his absence). Ibn Al-Qaasim was quoted to have said that it is permissible to conduct a marriage contract in such a case. Such possibilities are conceivable. The rule followed by the Muslim jurists in this regard is that if an evidence is open for (equally strong) possibilities, inference cannot be drawn from it (i.e. it cannot be held as incontestable evidence outweighing the opposing opinion).
As for the incident when the Prophet married off this woman, saying: “I have given you her in marriage for the part of the Quran which you know by heart,” Ibn Hajar explained it in Fat-h Al-Baari. Listing the deduced lessons from the Hadeeth, he said: “It is inferred from this Hadeeth that the ruler may act as Wali of a woman who has no Wali and give her in marriage to a man whom he believes to be suitable for her, provided that she gives her consent to it.” [End of Quote]
As for Umm Salamah her Wali was her son Salamah, as stated by Ibn Katheer in his book As-Seerah An-Nabawiyyah. He said, “The one who conducted her marriage contract was her son, Salamah ibn Abu Salamah, who was the eldest of her children.” [End of Quote] Salamah was an adult, unlike (her son) ‘Umar, who was a young boy at the time.
As for ‘Ali it was authentically reported on his authority that a marriage contract conducted without a Wali is invalid. Al-Bayhaqi cited the reports (you mentioned) in As-Sunan Al-Kubra and underlined that they were not authentic. Then he said: “Scholars held different opinions regarding the chain of narration of these reports and their texts. The common link in their chains of narration (i.e. a narrator from whom multiple chains of transmission emerge) was Abu Qays Al-Awdi, whose uprightness was controversial among scholars of Hadeeth. Also, the reliability of Bahriyyah, one of the narrators of the Hadeeth, is not verified… The first chain of narration on the authority of ‘Ali stipulating the Wali (as a condition for the validity of the marriage contract) is authentic and, thus, it is to be relied upon.” [End of Quote]
Regarding drinking water while standing up, the contradiction occurs in the indications of relevant authentic Ahaadeeth, which is not the case with the reports narrated on the authority of ‘Ali .
For more benefit on a woman’s marriage without guardian, please refer to Fataawa 92758, 85057, 226429, 132754, and 269080.
Allah knows best.