Ok so you've asked a ton of questions all at once, ill try to address them as best as I can. Also, it's been a while since i've played the game so bare with me.
It's unclear, though, how imports work – when does a ship decide to
Ok, so the game automatically EXPORTS your goods to other country's at a default price. But will never import anything unless you have 1 ship that will specifically get that item from a country and price you've set from your Trade Menu. If you need to import 3 goods, you must have 3 ships to do so, So 1 Dock is not going to be enough, you're going to need to build 2 docks to be able to do this.
Can only your trade ships import, or can your "export" ship also
Each ship you have, can only import or export, not both. As long as you have a dock, the goods you have produced on your island will automatically export like I've stated before. If you assign a specific ship you import coal for your power plant. You need to go to your trade menu, and do so, otherwise the power plant will never function.
Example, if you have 1 dock, which has 2 ships, You can have both ships importing Sugar and Coal, and your goods will still export at the default price.
Another Example. Lets say you're export 4 goods: Corn, Sugar, Coal, Oil. But only have 1 dock with 2 ships. And the default price on Corn is 800, and the default price on Oil is 6000. And your allies are paying 1200 for Corn, and 9000 for Oil. You setup those trade deals to make a better profit on Corn and Oil, But Sugar and Coal will still export anyway at their respective default price.
How much does a ship decide to import?
This is a very good question, And to be honest, I think the answer is "enough". I know it's a vague answer, but the way the Import and Export was setup in Tropico, was so you didn't have to stress over micromanaging EXACTLY how much of something you needed. So if you need uranium for your power plant and can't mine it, you'll obviously import it in. But lets say you decide you have 2 power plants that need uranium, and 1 power plant complains it's always low or out of uranium, then setup 2 ships to import it in instead of 1. Don't over think it.
And why are there trade routes for me to import goods at more than the
This could be for a number of reasons. The prices of trade goods are always fluctuating, and depending on what era of the game you're on. If 1 country at a certain time period has a monopoly on a certain good, then you'll have to suck it up and pay the price they are demanding. Supply and Demand. I mean, if you didn't have a lot of option to get goods from different countries at different prices, why bother having more than 1 source to import goods? it makes the game fun.
I had a Rum Factory create goods on a map where I had no Sugar
Plantation, but it did so in lumps of 1000 goods. Does this mean that
I'll only import exactly 1000 goods per ship, and only when the
factory is empty? Or can I get goods if the factory still has some
import goods, but has no other source of input goods?
Again, I never looked this in-depth with the Tropico's trading amounts, because the game really wasn't designed for this type of micromanaging, just import the good, it will come as needed and it will allow you to produce your finished products. If you have more than 1 Rum Factory, consider creating additional docks to import more sugar. Don't over think it. This isn't Sim City.
Does having additional Docks (and thus trade ships) increase the rate
at which I import goods?
Every time you create a new dock, you get 2 additional trade ships. This allows you to setup more deals in your Trade Section of the game. In that sense, yes you can increase the amount of goods you're importing if there are multiple sources of that good, But I would never do this. But no, it's not like you were importing X amount with 1 dock, and with adding another dock, you will start importing 25% more. This is False.
For example, lets say Your Allies are buying Oil for 9,000, and the Brazilians are selling Oil for 6,000. But you yourself do not produce any oil at all. How can you take advantage of this? Well Build 1 dock, And tell 1 ship to buy oil from the Brazilians and Sell it to your allies. You will make an easy 3,000 in the process for how ever many years you've placed your contract. So in this example, It has nothing to do with what you NEED to export and what you NEED to import, it's just about setting up trade deals to make you money. This is 1 of the ways having 3 or 4 docks can make your island rich.