My husband told me that he concluded our marriage with a fake name that belongs to another person, because he was sentenced. Nowadays, he recieved an ID, passport and birth certificate with the his current name. What is the ruling on being married to him?

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.  As regards being married to the person mentioned in your question: The marriage is valid so long as he was the same person your guardian concluded the marriage with, even if he changed his name i.e. if your marriage contract was concluded with the same person, since what counts regarding marriage is the persons not names. And Allah Knows Best.

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..

What should one who made an intention, at night, to make up for a missed fasting day, but broke his fast on the next day?

One who started making up a missed fasting day, then broke his/her fast without a legitimate reason is considered sinful, and is only obliged to make up for the missed fasting days. And Allah Knows Best.

What is the ruling on someone who perform their prayer after finishing it because they believe they missed a Rak'a, a prostration, or that they did not perform it correctly (thinking their prayer was invalid)?