Praise be to Allah.
A woman is obliged to cover her ‘awrah before those for whom it is not permissible to see it. The woman’s ‘awrah with regard to other women is the area between the navel and the knee, according to the majority of fuqaha’.
It is permissible to uncover the ‘awrah in cases of necessity, such as medical treatment, but that is conditional upon the treatment being absolutely necessary in the case of the most private part of the ‘awrah.
Based on that, there is nothing wrong with removing armpit hair with a laser, so long as that will not cause any harm.
With regard to having pubic hair removed by a female doctor, that is subject to the condition that there be a real need for that, such as if there is a lot of hair, and other methods such as plucking or shaving will not work, and you cannot remove it yourself by using a laser under the doctor’s instructions, so as to avoid her looking at your ‘awrah.
Al-‘Izz ibn ‘Abd al-Salaam (may Allaah have mercy on him) said: Covering the ‘awrah and pudenda is obligatory and is one of the best of customs, especially for non-mahram women. But it is permissible in cases of necessity and need. Examples of need include each spouse looking at the other, or doctors looking in the event that medical treatment is required. Examples of necessity include treating severe wounds. The condition of extreme necessity is stipulated with regard to looking at the pudenda that are not stipulated with regard to looking at any other part of the ‘awrah, because of the abhorrence of looking at it, and the conditions of necessity and urgency are stipulated with regard to looking at the pudenda of women that are not stipulated with regard to looking at the pudenda of men, because of the fear of fitnah that is connected to looking at women’s pudenda. By the same token, looking at the part of the thighs that is near the knees is not like looking at the buttocks. End quote from Qawaa’id al-Ahkaam (1/165).
Al-Sharbeeni al-Khateeb said: It should be noted that what is forbidden with regard to looking and touching is that for which there is no need. But in the case of need, looking and touching are permissible for cupping and treatment, even in the private area, because of extreme need, because prohibition in that case would cause a great deal of hardship. The Hanbalis regarded as a case where it becomes permissible to uncover the ‘awrah: shaving the pubic hair for one who is unable to shave it himself, as was mentioned by Ibn Muflih (may Allaah have mercy on him) in al-Furoo’ (5/153).
And Allaah knows best.