Is it permissible to recite the Holy Qur'an or to play a recording at mourning gatherings? Or is it considered an innovation?
Whenever Qur'anic verses are recited, it is obligatory upon everyone to listen, otherwise; they should remain silent. And Allah Knows Best.
Does Istigfar (Asking Allah for forgiveness) between the first and the second part of Friday prayer`s sermon render the prayer invalid?
Talking during Friday prayer`s sermon is disliked, but there is no harm in offering Istigfar between the two parts of the sermon, and it doesn`t invalidate the prayer as it is actually from Sunnah. And Allah Knows Best.
The supplication: "O Allah, we do not ask You to avert the decree, but we ask You for kindness in it." Is this Supplication Halal or Haram?
We are obliged to supplicate Allah, so when supplicating we comply with Allah's Ordainments, and making supplications that are reported in the authentic texts is better. And Allah Knows Best.
Is it permissible to move a grave from its current location to another place, for example, from a garden in a home to a cemetery, knowing that the grave has been there for a long time, about fifty years?
It is not permissible to move the deceased from his grave, as this may result in breaking his bones and violating his sanctity. Such actions are not allowed. And Allah Knows Best.