Is permissible to include a will requesting that a Holy Qur'an buried along with me?

It is not permissible to include a will requesting that a Quran be buried with the deceased in their grave. This is because the body of the deceased typically decomposes and may release impure substances, and it is not allowed to expose the Quran to impurity. Intentionally desecrating the Quran is an act of disbelief. Instead, read the Quran during your lifetime, as it will benefit you after death. And Allah Knows Best.

Is it permissible for a Muslim to slaughter an Aqeeqah on behalf of someone else, and offer it to him as a gift?

In principle, the guardian is the one who should offer the Aqeeqah (the sheep slaughtered on the seventh day from the child`s birth) because he is obliged to provide for the newborn, and it is impermissible for anyone else to slaughter it on his behalf unless with his consent. However, it is permissible for a person to offer the sheep, or its price as a gift to the guardian of the newborn, and then the latter can slaughter it, or deputies someone else to do that on his behalf.

Is supplicating in Qunoot, at times of affliction, during obligatory and voluntary prayer a Sunnah, and should it be done before Rukoo` (bowing), or after it?

At times of affliction, it is a Sunnah that Muslims supplicate in Qunoot after the final Rukoo` of each obligatory, or voluntary prayer as individuals, or in congregation.

A woman didn't fufill fasting of the month of Ramadan two years ago, due to pregnancy and breastfeeding, at the time being she is making up the missed Ramadan. What is the ruling of Sharia? And what is due on her?