Is washing the private part after urinating (Istinjaa`) a condition, and is it done with water and soap, or with water only?
Istinjaa` is obligatory for removing impurity, and it can be done with toilet paper, or a stone, or water. It is preferable to do Istinjaa` first with paper, or stone, or the like, then to wash the private part with water until making sure that the impurity has been removed. Using soap is not a condition, but there is no harm in doing so provided that it is washed off with water. It is also permissible to use either water, or paper if impurity is removed by any.
If a woman makes a vow to slaughter a sheep, and her husband is the one who buys it for her from his own money, and he says: "It is for you until you fulfill your vow with it," Is this permissible, or must she buy it herself from her own money?
If her husband gave her the sheep as a donation for the puprose of fulfilling the oath she made and was slaughtered by the wife or the husband on her behalf then the vow she made is fulfilled. And Allah Knows Best.
Is the `Iddah (waiting period) of the divorced effective from the time the divorce took place, or from the time of registering it at the court?
It is considered effective from the very moment her divorce took place, but not from the time of registering it at the court. And Allah Knows Best.
Is it permissible to offer mandatory prayer while sitting?
All perfect praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions. Standing up is a pillar in the mandatory prayer and the prayer is not valid without it unless one cannot do it. As for Nafila (supererogatory prayer), one can perform it in the state of sitting even if he/she can stand, but his/her reward is half the reward of the person who prays while standing if he/sh has no legal excuse for such an act. And Allah Knows Best.