What is the ruling on a person who broke the fast a day in Ramadan due to hardship?
He/she is obliged to make up that day.
Does touching one's wife invalidate ablution?
Touching the wife according to the Shafi'ee's doctrine invalidate husband's ablution. And Allah Knows Best.
What is the ruling on determining the sex of the baby whether through killing the female spermatozoa, or any other way in order to have a male?
If it is done through having a certain type of food, taking a certain type of medication, or organizing the time of the marital relation, then it is permissible.
Is it permissible to offer an absentee funeral prayer?
Yes, it is permissible to offer an absentee funeral prayer.