All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
There is evidence that it is not permissible for a woman to travel except with a Mahram (i.e. a husband or a person whom one is permanently prohibited from marrying due to suckling, marital, or blood relations); for more benefit in this regard, please refer to Fatwa 85945. In that Fatwa, we mentioned some exceptions which the scholars stated for cases of necessity.
However, it is not correct to use the narrations mentioned in the question to prove that it is permissible for a woman to travel without a Mahram for the following reasons:
The Hadeeth states the possibility of a woman travelling from Iraq to the Ka'bah while fearing no one but Allaah. Therefore, reporting this possibility does not mean that it is permissible to do so.
The travel of 'Aa'ishah and other mothers of the believers to Hajj does not necessarily mean that there was no Mahram with each of them, so this does not contradict the authentic narrations [about the prohibition of a woman travelling without a Mahram]. There might well have been a Mahram with them. Indeed it is confirmed that when the Prophet , prohibited a woman from travelling without a Mahram, he ordered one companion who informed him that his wife left for Hajj, to catch up with her and perform Hajj with her, despite the fact that the Companion was determined to go for Jihaad (to fight in the cause of Allaah).
Allaah Knows best.