Does Husband fasting while being impure for having sexual intercourse with his wife at night permissible?
Yes it is permissible for you to fast because impurity isn't among valid fasting conditions, but you are obliged to perform ritual impurity in order to catch up Fajr prayer at its due time. And Allah Knows Best.
If the imam does not leave time for reciting Surah Al-Fatihah, is the recitation of Al-Fatihah waived for the follower?
The recitation of Al-Fatihah is not waived for the follower. They should recite it silently, even if it causes a slight delay in following the imam in bowing (Ruku'). And Allah Knows Best.
Is it permissible for a Muslim physician to specialize in gynecology obstetrics, and what is the ruling on working as such?
Some fields of knowledge are an individual duty while others are a collective one, and specializing in gynecology obstetrics is a collective duty. However, if there were female physicians to treat women, then there is no need for a male gynecology obstetrics specialist to look at women`s Awrahs (private parts) except in necessary situations since the jurisprudential maxim says:” Necessity must only be assessed and answered proportionately.” And Allah Knows Best.
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.