It is established by textual proofs and consensus (ijmā’) that it is not obligatory upon a woman in her prayer to wear the jilbāb which covers her entirely when she is praying in her home. The jilbāb is only obligated when she leaves her home. So, she can pray in her home even if her face, hands and feet can be seen―as was the case when they would walk (in the pathways of Madinah) with the lesser covering before the command to cover completely was revealed. So, the parts of the body that must be concealed (i.e. the ‘awrah) in prayer have no connection to what must be concealed from the sights (of non-mahram men)―there is no relationship between the two.
(See Majmoo’ Al-Fatāwa of Shaikhul-Islām Ibn Taymiyyah, 22/115)
The allowance for the woman to pray with her feet uncovered in her home also seems to be the prefered opinion of Al-‘Allāmah Ibn ‘Uthaimeen (rahimahullāh) as he stated in Sharh Al-Mumti’ vol. 2, pp. 156-157.
As for the hadeeth where Umm Salamah (radiyallāhu ‘anhā) stated that she asked the Prophet (salallāhu ‘alaihi wasallam): “Can a woman pray in a long dress and scarf without wearing a lower garment (izār)?” He replied: “Yes if the dress is ample and covers the surface of her feet.” (Abu Dawūd, no. 640) This hadeeth was graded weak (da’eef) by Al-Imām Al-Albāni.
Note: There is some differing among the scholars in the issue of whether a woman needs to cover her feet when praying among women and male mahrams. Al-Imām Ibn Bāz (rahimahullāh), for example, holds that the feet of a woman must be completely covered in prayer, even when she is praying alone.
As for what a woman must wear in public, then please read this important article on Hijāb: Understanding the Muslim Dress Code: Modesty for Men and the Hijāb for Women