I apologize in advance if my suggestions are too obvious, or if you have already tried them. Since you said that you are out of ideas, here's some of mine.

First, hold down the V key and look at hte sea regions. Those that are green should support trade. Red regions are blocked.

Second, instead of clicking on your port, click on the gold coins icon in the lower left. This brings up the province economy screen (yes I know you already knew that - I don't think you're stoopid). Click on the name of the province that you think should be producing trade revenue. This will open up sub-screen showing trade for that province.

I think that it is not the port that earns revenue, its the Merchant.

And no, you dont need a special kind of ship. All ships are warships; they dont carry trade goods, they protect shipping in the sea region.

Hope this helps. Good luck mate

ichi