What is the ruling of Islamic Law on following the actions of the imam in prayer and how this following is achieved?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
The follower (ma'mum) is required to follow his imam in the actions of the prayer. This "following" (mutaba'ah) means that the follower performs each action of the prayer after the imam has begun it but before he has finished it. For example, the follower bows (in ruku') after the imam has reached the position of bowing, then rises after the imam has risen. The Prophet, peace and blessings be upon him, said: "The imam is only appointed to be followed, so when he says the takbir, say the takbir; and when he bows, then bow; and when he prostrates, then prostrate" (agreed upon, i.e., reported by both al-Bukhari and Muslim).
It is stated in Mughni al-Muhtaj (1/505): "Among the conditions of valid congregational prayer (iqtida') is following the imam in the actions of the prayer... meaning that following the imam is obligatory in the physical actions of the prayer, not in its verbal utterances... Complete following (kamal al-mutaba'ah) is achieved when the follower's beginning of an action comes after the imam's beginning of that same action, while the follower's beginning of the action precedes the imam's completion of it" — end of quote, with slight paraphrasing.
And Allah, the Exalted, knows best.
Does vomiting during the day in Ramadan break the fast?
Intentional vomiting is one of the nullifiers of fasting; whoever vomits deliberately breaks their fast.
However, if vomiting occurs involuntarily, the fast remains valid as long as nothing returns to the body cavity (jauf). If anything is swallowed back, the fast is invalidated.
The Prophet ﷺ said: "Whoever is overcome by vomiting does not have to make up the fast, but whoever induces vomiting deliberately must make it up." [Narrated by Abu Dawood and At-Tirmidhi]
What is the ruling on performing dry ablution (tayammum) instead of the ritual bath due to severe cold?
Whoever is sick such that he cannot use water, or finds the water cold and heating it is not possible, and the person would be harmed by using cold water, it is permissible for him to perform tayammum. He must make up what he prayed with tayammum, because this is a place where there is water. And Allah the Almighty 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.