Is it permissible for one to shake hands with his uncle`s daughter?
No, it isn`t because he is a non-Mahram (Marriageable) to her.
What is the ruling on performing Tahajjud after the Witr?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is permissible for one who has prayed the Witr to perform voluntary (Nafl) prayers after it. However, it is preferable for the Witr to be the final prayer of the night. Therefore, if a person is confident that they will wake up during the night for Tahajjud, it is recommended for them to delay the Witr until after the Tahajjud. Conversely, if one fears they may not wake up, they should perform the Witr before sleeping.
Al-Khatib al-Shirbini (may Allah have mercy on him) stated: 'It is not disliked (Makruh) to perform Tahajjud after the Witr, but it is not recommended to do so intentionally.' [Mughni al-Muhtaj, Vol. 1/P.454]. And Allah the Exalted knows best."
What is the ruling on entering the toilet with something containing the remembrance of Allah?
It is disliked to bring into the toilet anything containing a reminder of Allah, such as the Quran or a sacred name (like the name of Allah, Muhammad, etc.). If one does that, etiquette dictates that he conceal it so it is not visible while entering the toilet, either by cupping his hands over it, putting it in his pocket, or inside his shirt if it is a necklace or ring. And Allah the Almighty knows best.
Is it enough to say Tasbeeh (Saying Subhaana Rabbiya Al-‘Atheem during Rukoo`, and saying Subhaana Rabbiya Al-A‘laa during Sujood ) once, or twice during Rukoo`(Bowing in prayer) and Sujood (Prostration)?
All perfect praise be to Allah, The Lord of The Worlds Yes, saying Tasbeeh once during Rukoo` and Sujood is enough, provided that the worshiper`s attentiveness of the heart isn`t undermined during each. As regards the recompense, it depends on the number of Tasbeeh. And Allah Knows Best.