Finding a Spy: Two options: Either he/she is walking around town, and you have to go around talking to random people until you find the right one, or they're hiding inside a house. Both of those are really easy, from what I've seen.
Killing the Pirate: Frankly, this is probably the one I've had the most trouble with so far, but it is extremely straight-forward.
Pick up letter in enemy town: Either with a high sneaking skill, moor at the town at night, or sneak through the gates, after mooring at a beach and making your way through the jungle. Once inside the town, go to the inn and rest until morning. The town guard don't care much for you at day, unless you walk straight towards them or something (depends on your sneak skill, I figure. Mine is low). Now, the Governer should have told you who to find (etc. Harbor Master, Shopkeeper, Shipyard-man, or moneylender are the ones I've had), so just go talk to them. They'll tell you not to contact them again, blah-blah. Once you have the package, make a run for your boat/the gates (if you moored at the town, you might want to continue to avoid the guards. But if you moored outside, it doesn't really matter).
Kill Outlaw: Just walk through the jungle until a man comes up to you and tries to steal your money. You talk a bit about the Guv'ner, until you decide to walk away or attack him (you can walk away first, then stab him in the back too!). Easy.
Deal with smugglers: You have to have contraband on your ship, be moored at town, and talk to the smuggler in the inn. He'll tell you where to meet. Go there, and attempt to arrest them, and you'll have to fight and kill all three of them.
Those are the missions I've gotten from the governor so far. As for rank: Yes, I've managed to go one up in rank, and get a card-thing so I can get missions from the Governor-General. Warning! Those missions are extremely tough! When I got the card, I also got friendlier with the nation (France, in my case).