Is it permissible for a young man and a young woman to enter into a covenant as a married couple without having a true seclusion (Khalwah) until their marriage contract is officially concluded?

The relation which isn`t based on a marriage contract is forbidden, and entering into a covenant as a married couple is unaccounted for and is from the evil suggestions of the devil.

Is Iddah (waiting period) incumbent on an old woman whose husband had died?

Upon the death of the husband, Iddah is incumbent on the wife regardless of their age. And Allah Knows Best.

Generally speaking, I`m dutiful to my mother and opt for obeying her. However, she chose a certain girl to be my future wife, but I`m not pleased with her choice although that girl comes from a good family and enjoys good character. Should I obey my mother or choose another girl that lives up to my expectations?

Praise be to Allah the Lord of the Worlds.

One must obey their parents in whatever is in compliance with the teachings of Sharia and within one`s capacity. However, you aren`t to blame for not marrying that girl, but it is better that you tell your mother about that and it is preferable that you perform Istikharah prayer. And Allah The Almighty Knows Best.

Is it permissible for a woman to visit the doctor while in her `Iddah (waiting period)?

It is permissible for her to do that during day time, but it is impermissible for her to leave her house at night except for a sound reason. And Allah Knows Best.