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).

What is the expiation for perjury?

One who commits perjury should repent, seek Allah`s forgiveness and offer an expiation which is: feeding ten indigent persons, or clothing them, or giving a slave his freedom, but if that is beyond his means then, he should fast for three days. Allah, The Almighty, Says in this regard (What means): "Allah will not call you to account for what is futile in your oaths, but He will call you to account for your deliberate oaths: for expiation, feed ten indigent persons, on a scale of the average for the food of your families; or clothe them; or give a slave his freedom. If that is beyond your means, fast for three days. That is the expiation for the oaths ye have sworn. But keep to your oaths. Thus doth Allah make clear to you His signs, that ye may be grateful." [Al-Ma`idah/89].

Is it permissible to donate corneas to Jordan Eye Bank or all the organs via telephone?

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.

Corneas donation that is carried out after one's death is permissible by the Board of Iftaa'. Because this procedure is highly sensitive, the donation process

should undergoes to an accredited mechanism. And Allah, The Exalted, Knows Best  

Should water be put in the deceased`s mouth and nose while washing him/her?

Putting water in the mouth and nose of the deceased while washing him is impermissible.