Is it valid to perform Tarawih prayer at home?
It is Sunnah for a Muslim to perform Tarawih prayer in congregation at the mosque.
However, if a person sometimes prays it at home in congregation with their family for a valid reason, there is no harm in doing so.
What is the ruling on X-ray imaging while fasting in Ramadan?
X-ray imaging itself does not break the fast. However, if a person ingests a contrast agent or medication to enhance the imaging process through an open passage to the body cavity (jauf)—such as the mouth or rectum—then their fast is invalidated.
Is it permissible for the young woman to turn the suitor down if she had no feelings for him?
Yes, it is permissible for her to do so.
What is the ruling on someone who is uncertain whether he omitted one of the ab‘āḍ of the prayer, such as being uncertain whether he omitted the first tashahhud?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
The ab‘āḍ include the middle tashahhud and the qunūt in Fajr prayer — for which sujūd as-sahw (the prostration of forgetfulness) is prescribed if left out. Whoever is uncertain, during the prayer, whether he omitted one of these elements, should proceed on the assumption that he did not perform it, and should perform sujūd as-sahw accordingly.
It is stated in ‘Umdat as-Sālik (p. 63): "If he is uncertain... whether he omitted a specific ab‘āḍ element, or whether he performed sujūd as-sahw, or whether he prayed three or four rak‘ahs — he proceeds on the assumption that he did not do it, and performs sujūd as-sahw."
The original principle (al-aṣl) for someone uncertain about [having performed] the first tashahhud is that he assumes he did not perform it. And Allah, the Exalted, knows best.