Is a woman`s face an Awrah (a private part) that should be covered?
During prayer, a woman`s whole body is a private part except the hands and the face. Outside the time of prayer, scholars are of the view that a woman should wear on her Islamic attire to avoid temptation.
Is it permissible for the man to go out wearing shorts?
All Perfect Praise is due to Allah, The Lord of The Worlds
It is permissible so long as his Awrah (private part), which is between his navel and knee, is covered. However, the regular wear traditions and modesty are to be observed. 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).
I took a car from the Orphans' Fund Development Institution; is there usury (Riba) involved in this transaction?
If the Sharia's conditions were met, then there is no harm in this. And Allah The Exalted Knows Best.