Is it permissible for the woman to use her mobile phone while on the street?
All Perfect Praise is due to Allah, The Lord of The Worlds
Modesty should be observed by the woman at all times, so she is free to use her phone while on the street as long as it is done within the limits of Islamic manners. And Allah The Almigty Knows Best.
Is it permissible for someone who has forgotten to mention the supplication pertaining to answering the call of nature to say it while in the toilet?
It is disliked for a Muslim to mention Allah while in the toilet, but he/she may recall the supplication in his/her head without uttering it; that is in case he/she had forgotten mentioning it before entering the toilet.
An Imam stood to offer a fifth Rak`ah in a four-Rak`ah prayer, and he was reminded to sit by those behind him, but he wouldn`t listen. What is the ruling on those who followed his lead knowingly and intentionally?
If the Imam stood to offer a fifth Rak`ah, those praying behind him shouldn`t have approved of that, and the prayer of those who did is considered null and void.
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.