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.
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
Is keeping money in and taking profits from the Islamic Bank lawful?
If Allah Wills this is lawful, because they stipulate on theselves to work in accordance to the guidlines of Islamic Law. And Alalh Knows Best.
Is it permissible to listen to chants praising the Prophet if accompanied by musical instruments?
If singing, or chanting in praise of the Prophet was accompanied by a forbidden musical instrument, then it is forbidden.