Is it permissible for a woman to go out wearing bright colors?

Bright colors are part of a woman`s adornment, and it is impermissible for her to show it to people other than her husband, or one of her Mahrams (Unmarriageable kin).

Does the deceased hear the supplications or recitations of those visiting his/her grave? And does making constant supplications for the deceased who didn`t adhere to performing the prescribed prayers ameliorate his/her affliction?

Praise be to Allah the Lord of the Worlds.

In principle, the dead hear the living since it has been rigorously authenticated that the Prophet (PBUH) ordered the bodies of the idolaters slain in the Battle of Badr to be buried. He then approached them and called them one by one and said: "Have you found what your Lord promised to be true for we have found what our Lord promised us to be true." Umar asked him: "O messenger of God! Why do you speak to lifeless bodies?" The Prophet (PBUH) replied: "By the One Who has sent me with the truth! You do not hear my words better than they do except that they cannot respond." [Agreed upon].

However, not all the deceased are in the same situation because Allah The Almighty Knows: "Before them is a Partition till the Day they are raised up." [Al-Mu`minun/100]. Moreover, supplication reaches the deceased. And Allah 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 go out at night except for a sound reason.

Is the `Iddah (waiting period) of the divorced effective from the time the divorce took place, or from the time of registering it at the court?

It is considered effective from the time the divorce took place, and not from the time of registering it at the court.