Is it permissible, in Sharia, to take profits from an Islamic bank, given that a partnership contract for profit and loss has been signed, and am I obligated to pay zakat again if the bank handles its distribution?

The contract between the client and the Islamic bank is a Mudarabah contract, in which both parties share the profit and the capital bears the loss. It is permissible for the client to take his share of the profit. If the bank undertakes to pay Zakat on behalf of the client, then the client is not obligated to pay it again.