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.
Is it permissible for a menopausal woman who has blood discharges to fast?
If she reached the age of menopause, and her menses stopped, but later she saw blood discharges that lasted a complete day and night (24) hours, then it is menstrual blood. Still, if these blood discharges lasted less than a complete day and night (24) hours, then it is not menstrual blood, but it is bleeding outside the monthly period (Istihadah), and it does not prevent her from prayer and fast, but she has to make ablution for every prayer.
Is it permissible to watch soap operas during leisure time?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
The answer depends on what is shown in them. If they include what is forbidden to see or hear then they aren`t permissible to watch. And Allah The Almighty Knows Best.
Is it permissible to offer an absentee funeral prayer?
Yes, it is permissible to offer an absentee funeral prayer.