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 to slaughter one animal as a sacrifice and an Aqeeqah (the sheep slaughtered on the seventh day from the child`s birth)?
It is impermissible to do so since each of them is slaughtered for a different reason.
Is it permissible to swear by (Allah ”The Living”)?
Yes, it is permissible because “ The Living “ is one of Allah`s Attributes.
Are phone conversations between the two sexes permissible if for consultation, and within the limits of good manners?
Such act is impermissible since it renders hearts corrupt, and could lead to forbidden attachment. Allah, The Almighty, says: “and follow not the footsteps of the devil. “{Al-Baqarah/68}.