What is the ruling on someone who vowed to fast for two consecutive months, but failed to do so?
He is obliged to fulfill his vow by fasting for two consecutive months, but if he fails to do so then, it remains a debt on him until he either fulfills his vow, or dies. In case of his death, his guardian is supposed to fast on his behalf, or donate food from his estate in return for fasting, but if the guardian fails to do so then, it is all in the Hands of Allah.
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 it permissible to pray with a heater (stove) in front of me?
There is no harm in praying with a heater in front of you, as long as you do not intend to venerate the fire or imitate those who worship it. 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.
