Is it permissible to swear by (Allah ”The Living”)?

Yes, it is permissible because “ The Living “ is one of Allah`s Attributes.

Is it disliked to clip one`s nails at night, and is it permissible to burn them afterwards?

It is desirable to bury the cut hair and the clipped nails. As for burning clipped nails, there is no evidence forbidding that , but it is better to suffice with the aforesaid ruling-burying cut hair and clipped nails-as it is based on a well known evidence.

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 to shorten the eyebrows if they get too long?

Yes, it is permissible, but not desirable.