Is a woman`s neck an Awrah (private part)?

All Perfect Praise is due to The Lord of The Worlds.

It is an Awrah, and it is forbidden for her to reveal it before non-Mahrams (Marriageable men). And Allah Knows Best.

What is the ruling when the monthly period when exceeding (15) days?

The maximum duration of menstruation is 15 days. If it exceeds this, it is considered irregular bleeding (Istihada). In this case, one must perform ghusl (Ritual bath) after menstruation, pray, and make wudu for each prayer after its time begins. She should return to her previous menstrual habit regarding duration and timing. For example, if her period used to be from the 20th to the 27th of each month, this is considered her menstruation period. Anything before or after this is Istihada, during which she only leaves prayer and fasting for the habitual menstrual days, then performs ghusl and resumes praying.

How to pay Zakah (obligatory charity) due on articles of merchandise?

Articles of merchandise are estimated by their whole sale market price at the end of each lunar year, and (2.5%) of their value is paid as Zakah whether it (value) went up, or down compared to actual purchasing price, and whether the increase (profit) was in the article itself such as an increase in the animal`s weight, or in the prices themselves. 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).