I used artificial nails on my wedding night, but I was surprised to find out that they require at least a week to remove. However, I performed all my prayers during that period. What is the ruling on those prayers?
If these nails prevent water from reaching one's skin then your prayer are invalid and obliged to redo all of them. And Allah Knows Best.
Is it permissible for the guardian, or the father of the child to hit the latter in order to teach him/her good manners?
It is permissible for the guardian to hit his/her child for teaching them good manners once they become ten years old provided that it is done mildly, harmlessly and without intimidation. Actually, such punishment is to be employed gradually i.e. after: asking, warning, and scolding. As for the teacher, he isn`t allowed to hit the student unless the latter`s father approves of it, and it is done according to necessity without intimidation, or harm.
Is hair extension permissible?
All Perfect Praise is due to Allah, The Lord of The Worlds
It is impermissible for the man and the woman to apply a human`s hair as extension, but using artificial hair is permissible for the married woman after obtaining the approval of her husband so long as non-Mahrams don`t see her. And Allah Knows Best.
In the last verse of Surah At-Tin, Allah Says (What means): "Is Allah not the most just of judges?" [At-Tin/8]. Is it permissible for us, while performing congregational prayer, to respond after the Imam recites it by saying: "Indeed, and I am among those who bear witness to that." ?
It is recommended for the worshipper to say 'Indeed' (Bala) silently after reciting or hearing the verse (/What means): "Is Allah not the most just of judges?" [At-Tin/8]. This is part of reflecting on the Quran during its recitation or listening, as Allah Says (What means): "Then do they not reflect upon the Quran, or are there locks upon [their] hearts?" (Muhammad: 24). Doing so helps in achieving humility and focus in prayer. And Allah Knows Best