What is the ruling on mentioning Allah`s name upon slaughtering an animal?
In the Name of Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is a Sunnah (prophetic tradition) for the person slaughtering to say: "Bismillah" (In the Name of Allah), though the most complete and perfect phrasing is to say: "Bismillah ar-Rahman ar-Rahim" (In the Name of Allah, the Most Gracious, the Most Merciful). This is in accordance with the words of Allah Almighty: "So eat of that [meat] upon which the name of Allah has been mentioned." (Al-An'am/118)
However, pronouncing it is not an obligatory requirement (Wajib). If someone omits it—whether deliberately or out of forgetfulness—the meat remains entirely lawful (Halal) to eat, though purposely omitting it is considered disliked (Makruh). And Allah the Almighty Knows Best.
What is the ruling on objecting to the Mosque Imam regarding the joining of prayers due to rain?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is not permissible for any of the congregants to object to the Imam in the mosque, whether the objection is to demand the joining of prayers or to oppose it; for the general principle is that the Imam possesses the religious knowledge (Al-Ilm al-Shar‘i) that qualifies him to make the decision in this matter. And Allah the Exalted knows best."
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 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.