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.
What is the ruling on someone who prays Fajr late, after sunrise, knowing that they wake up on time but return to sleep and do not perform it within its time?
It is obligatory to perform the prayer within its prescribed time. If someone wakes up and knows how to act, they must perform it before the time ends. If they delay it, they are sinful and must make it up. And Allah Knows Best.
Is it permissible for a person to sever ties with his sister because her husband had quit prayer and done him harm?
A sister is a blood-relative and maintaining blood-relations is obligatory in Islam. As regards her husband, he should be invited to the way of Allah with wisdom and beautiful preaching.
During a flight from Amman to the UAE, while we were flying over Saudi Arabia, the captain announced that it was time to break our fast, coinciding with the exact time of sunset. After I had broken my fast with my first date, the captain apologized for the earlier announcement and clarified that the correct time would be in 20 minutes. Am I required to continue fasting on that day?
Whosoever break his fast at teh very first announcement to make up the missed day of Ramadan, since Allah, The Most Exalted, Said (What means): "then complete your fast Till the night appears" [Al-Baqara/187]. And Allah Knows Best.