What is the difference between Tahajjud prayer and night prayer, and do they have a particular Witr prayer other than that of Isha (evening prayer)?
Tahajjud and Qiyam-Al-Lail (night prayer) are two words for the same meaning which is offering voluntary prayer at night after sunset, but before dawn break. However, Tahajjud is offered after waking up; whereas, Qiyam-Al-Lail is permissible before sleeping, or after it. Therefore, every Tahajjud is a Qiyam, but not every Qiyam is a Tahajjud. Moreover, there is no specified Witr for both of them, but Witr after Isha is by itself the Witr of Qiyam, and it is a Sunnah to postpone it until after Qiyam if the worshiper thought that he was most probably going to wake up in order to offer it, but if he wasn`t sure of waking up, then he is at liberty to offer it before going to bed as was reported in the Prophet`s Hadith.
Is Iddah (waiting period) incumbent on an old woman whose husband had died?
Upon the death of the husband, Iddah is incumbent on the wife regardless of their age. And Allah Knows Best.
Someone asked me to pay off his debt on his behalf as a loan, without any compensation. When I went to the creditor, he told me that if I paid the full amount at once, rather than in installments, he would give me a certain discount. Is this permissible? And if he applies the discount, to whom does the deducted amount belong?
If part of the debt is paid and the creditor forgives the remaining amount, the waiver is valid, and the remaining debt is no longer the responsibility of the original debtor. The person who paid the debt on behalf of another has no right to claim any portion of the original debt. And Allah Knows Best.
What is the ruling in Islam on castrating animals to use them for work?
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.
This isn`t permissible because jurists said: "It is permissible to castrate the animal if it is eaten and this procedure is done at early age." And Allah The Almighty Knows Best.