I know you can exterminate a population with the "Black Death" card.

Is there any other methods of exterminating a port population?

Not yet. Black Death voodoo is the only way. You can buy cards at the market.

Tis' the most evil, vile card out there. Steer clear of black death mate... nothing good comes of it.

It is the only way to accomplish nation missions. Correct?

You will need multiple cards to achieve the goal also.