Is it permissible to deal with brokerage firms for trading permissible stocks on the stock exchange, knowing that these firms deal with both permissible and impermissible stocks?

There is no harm in dealing with a broker to buy permissible stocks that you specify for him. The Prophet, peace and blessings be upon him, and his companions dealt with the Jews despite their involvement in usury, and that did not prevent them from dealing with them in what is permissible.