What is the ruling on having brotherly ties between a strange man and a strange woman, and is the former considered a Mahram (unmarriageable) to the latter?
Such a relation between a strange man and a strange woman is forbidden in Islam because it involves forbidding what Allah has allowed by marriage, and allowing what Allah has forbidden such as looking and the like. Such a relation doesn`t render any act lawful between them, thus they are forbidden to look at each other, have a Khulwah (seclusion), and travel together.
What is the ruling on a person who isn`t able to fast due to old age, or an incurable disease?
Paying the ransom is due on such a person, and that is giving a Mud (600 grams) of wheat, or rice, or the price of that amount to a needy Muslim for each missed fasting day. And Allah Knows Best.
Is it permissible to repeat the Istikhara (seeking guidance) prayer, and is it a condition that one should sleep after offering it?
All perfect praise be to Allah,The Lord of The Worlds Istikhara is offered once, but it is permissible to repeat it, and sleeping after offering it is not a condition for its validity. And Allah Knows Best.
Is it permissible for the children of a deceased father to settle his debt from the Zakah (obligatory charity) money due on them?
It is impermissible to use the Zakah of one`s money for settling the debts of the deceased. However, children should settle the debts of their deceased parents out of filial piety. And Allah Knows Best.