Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.
What should a person who doubted having missed a pillar of prayer do?
All praise be to Allah, The Lord of The Worlds. The prayer of one who doubts having missed one of its pillars is valid, and nothing is due on him/her since having a doubt after offering the act of worship doesn`t undermine its validity. And Allah Knows Best.
What is the Ruling on Reciting the Prophetic Narration?
All Perfect Praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.
Reciting the Prophetic narration is permissible as many scholars used to, since in principle, the rulings of recitation have to be observed while speaking Arabic language. And Allah Knows Best.
Is it permissible for the doctor to uncover the hair of a female patient out of necessity?
It is only permissible for him to do so when there is no female doctor, and he should examine the patient in the presence of one of her Mahrams (unmarriageable kin).