It is permissible for the Qazi (judge) and the witness to see the face of the woman. In the same way, it is permissible to see the face of the woman whom one desires to marry. There is a Hadith in Jame' Tirmidhi:
Translation of Hadith: The Holy Prophet (Sallallahu alaihi wa sallam) said: See the face of the woman whom you want to marry as this is a means of maintaining love. (Jame' Tirmidhi, Hadith No. 1007)
At the time of proposing marriage, it is allowed only to see the face and the hand to the extent of the palms. Seeing the face it is to get to know the facial features and the beauty and seeing the hands is to judge the condition of the body parts. If more detail is desired, any reliable lady might be sent to know the other details about the girl. In the same way, if the girl wants to see the boy for the purpose of marriage, the Shariah allows it.
It is given in Radd Ul Muhtaar, Kitab Ul Hazr.
It is permissible for the doctor and/or the nurse to see the spot of illness/ailment of the patient. Those parts of the woman's body, which are prohibited to be seen, they remain prohibited by the Shariah even if they are separated. This is the same ruling for women's hair and nails of her legs.
Those parts of the body of woman which are prohibited to be seen, it is also strictly prohibited to touch or caress them.
The Holy Prophet (Sallallahu alaihi wa sallam) has said:
Translation of Hadith: Whoever touches the palm of a woman, with whom he does not have a valid relationship, in the Qiyamah, fire will be placed on his palms. (Takmila Fath Ul Qadeer Hidayah Kitab Ul Karahiyyah, Vol. 4, Pg. No. 459)
The Holy Prophet (Sallallahu alaihi wa sallam) has also said:
No man should see the Satar (the area from below the navel to the knees) of another man and no woman should see the Satar of another woman (Sahih Muslim, Hadith No. 512)
In Musannaf Abdur Razzaq, the saying of Hadhrat Umar (May Allah Ta'ala be well pleased with him) is mentioned: It has been narrated on the authority of Hadhrat Suleiman bin Mas'har (May Allah Ta'ala be well pleased with him) that Hadhrat Umar (May Allah Ta'ala be well pleased with him) said: You do not let your women wear very fine clothes, as they will be in such a state that their body parts will be visible through it. (Musannaf Abdur Razzaq, Hadith No. 12142)
Burqa is the safest means to maintain chastity. The Burqa and its nose piece fulfill the same objectives, for which the command to cover the whole body with a chador, including the face, was given.