Is it permissible for a Muslim woman to go out wearing adornment and not putting on her Hijjab(Islamic wear)?

It is impermissible for a Muslim woman to go out wearing adornment, and not having her Hijjab on.

Is it permissible to fast on behalf of the dead who died owing missed fast to make up ?

His relative should fast on his behalf, and it is permissible for the guardians to give permission to non-relatives of the dead to fast on his behalf as well.

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 (peace and blessings be upon him) 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 (peace and blessings be upon him) 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.”{Related by Bukhari & Muslim}.

However, not all the deceased are in the same situation because Allah the Almighty says, "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 shake hands with her brother-in-law?

It is forbidden for the woman to shake hands with her brother-in-law because he isn`t a Mahram(unmarriageable) of hers.