What is the ruling on opening a savings account with profits at Abu Dhabi Islamic Bank Egypt, given trust in its Sharia board, and is it obligatory to monitor all the bank's operations, or is it permissible to rely solely on trust in the board?

The basic principle regarding transactions of Muslims and Islamic financial institutions is validity and permissibility. Their dealings are শরعية (religiously lawful) unless proven otherwise. If doubt arises concerning a specific transaction, the matter must be referred to the bank's Sharia Board.