Are the dealings and profits of Faisal Islamic Bank of Egypt permissible, given that it is subject to the Central Bank of Egypt, which relies on usurious transactions?

If there is a pre-determined and unchangeable profit margin between you and the bank, this is usury (riba). However, if the bank invests customers' money in permissible ventures without specifying a fixed amount for profits, then this is a legitimate Mudarabah (profit-sharing) contract. There are many permissible investment avenues not limited to banks.